The (11)/(6) in decimal form is 11 ÷ 6 = 1.8333333…
To convert 11/6 into decimal form, divide 11 by 6. 11 ÷ 6 = 1.8333333…
To indicate which digit or group of digits repeat, we can put a bar above the repeating digits.
The repeating digits start immediately after the decimal point.
Therefore, the decimal representation of 11/6 is 1.83 with a bar above the digit 3.
How to convert a fraction to a decimal?
To convert a fraction to a decimal, we have to divide the numerator (top number) by the denominator (bottom number). This method will work for any fraction, whether it is a proper fraction (numerator is less than the denominator), an improper fraction (numerator is greater than or equal to the denominator), or a mixed number (a whole number and a fraction).
Dividing Fractions: To divide fractions, we have to multiply the numerator of the first fraction by the denominator of the second fraction and multiply the denominator of the first fraction by the numerator of the second fraction. Then, simplify the fraction if necessary. The resulting fraction will be the quotient of the two fractions.
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if using a confidence interval, you must state the point estimate, the margin of error, the interval, and the conclusion;
if using a hypothesis test, you must state the hypotheses, the test statistic, the p-value or critical value, and the conclusion;
if using regression, you must state the correlation coefficient, the regression equation (even if it is not a good model), any predicted values, and the conclusion.
If using a **confidence interval**, you must state the:
- **Point estimate**: The estimated value of the parameter based on the sample data.
- **Margin of error**: The maximum amount of error expected in the estimate.
- **Interval**: The range of values within which the true parameter value is likely to fall.
- **Conclusion**: A statement about the parameter based on the interval.
For example, if estimating the mean height of a population with a 95% confidence interval, you might state:
"The point estimate for the mean height is 170 cm, with a margin of error of 2 cm. The 95% confidence interval for the mean height is (168 cm, 172 cm). We can be 95% confident that the true mean height falls within this interval."
If using a **hypothesis test**, you must state the:
- **Hypotheses**: The null and alternative hypotheses, which represent different assumptions about the population parameter.
- **Test statistic**: The calculated value used to assess the evidence against the null hypothesis.
- **P-value or critical value**: The probability of observing the test statistic or a more extreme value, or the critical value beyond which the null hypothesis is rejected.
- **Conclusion**: A decision to either reject or fail to reject the null hypothesis based on the test results.
For example, if testing whether a new treatment is effective, you might state:
"The null hypothesis is that the new treatment has no effect, while the alternative hypothesis is that it does have an effect. Based on the calculated test statistic of 2.18 and a significance level of 0.05, the p-value is 0.030, which is less than the significance level. Therefore, we reject the null hypothesis and conclude that the new treatment has a statistically significant effect."
If using **regression analysis**, you must state the:
- **Correlation coefficient**: A measure of the strength and direction of the linear relationship between the variables.
- **Regression equation**: The equation that describes the relationship between the dependent and independent variables.
- **Predicted values**: Values estimated by plugging in independent variable values into the regression equation.
- **Conclusion**: A statement about the relationship between the variables based on the regression analysis.
For example, if analyzing the relationship between sales and advertising expenditure, you might state:
"The correlation coefficient between sales and advertising expenditure is 0.80, indicating a strong positive relationship. The regression equation is Sales = 1000 + 0.5 * Advertising, where Sales represents the predicted sales value based on the advertising expenditure. Using this equation, we can estimate the sales value for a given advertising expenditure. However, please note that this regression model may not be a good fit based on the coefficient of determination (R-squared) of 0.60, suggesting that 60% of the variation in sales is explained by the advertising expenditure."
Remember to adapt the specifics to your actual analysis and results.
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Section 1.5
18. If $10 is invested for 15 years at 3% interest compounded continuously, find the amount of money at the end of 15 years. Answer correct to one decimal place. 19. Evaluate log4 32 20. Find the domain of the function g(x) = log3(3-3x)
21. Solve the equation 3x2+2 = 27x+4
22. Solve the equation log5 (2x-1)-log5 (x-2)= 1
18. The formula for calculating the amount of money accumulated with continuous compounding is given by the formula:
A = P * e^(rt),
where A is the amount of money at the end of the investment period, P is the principal amount (initial investment), e is the base of the natural logarithm (approximately 2.71828), r is the interest rate, and t is the time period in years.
In this case, P = $10, r = 3% (or 0.03 as a decimal), and t = 15 years. Plugging in these values into the formula, we have:
A = 10 * e^(0.03 * 15).
Using a calculator or computer software, we can calculate this as:
A ≈ 10 * 2.22554.
Rounding to one decimal place, the amount of money at the end of 15 years is approximately $22.3.
19. To evaluate log4 32, we need to determine the exponent to which 4 must be raised to obtain 32. In other words, we want to solve the equation:
4^x = 32.
Taking the logarithm of both sides with base 4, we have:
log4 (4^x) = log4 32.
Using the property of logarithms that states log_b (b^x) = x, the equation simplifies to:
x = log4 32.
Using a calculator or computer software, we can evaluate this as:
x ≈ 2.5.
Therefore, log4 32 is approximately equal to 2.5.
20. The domain of the function g(x) = log3(3-3x) is determined by the argument of the logarithm. For the logarithm to be defined, the argument (3-3x) must be greater than zero. So, we need to solve the inequality:
3 - 3x > 0.
Simplifying this inequality, we have:
-3x > -3,
x < 1.
Therefore, the domain of the function g(x) is all real numbers less than 1.
21. To solve the equation 3x^2 + 2 = 27x + 4, we need to gather all the terms on one side and set the equation equal to zero:
3x^2 - 27x + 2 - 4 = 0,
3x^2 - 27x - 2 = 0.
Now, we can solve this quadratic equation by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients of the quadratic equation (ax^2 + bx + c = 0).
In this case, a = 3, b = -27, and c = -2. Substituting these values into the quadratic formula, we have:
x = (-(-27) ± √((-27)^2 - 4 * 3 * (-2))) / (2 * 3),
x = (27 ± √(729 + 24)) / 6,
x = (27 ± √753) / 6.
Therefore, the solutions to the equation are:
x ≈ 1.786 and x ≈ -5.786 (rounded to three decimal places).
22. To solve the equation log5 (2x - 1) - log5 (x - 2) = 1, we can use the properties of logarithms. The subtraction of logarithms is equivalent to the division of their arguments. Applying this property, we have:
log5 ((2x - 1)/(x
- 2)) = 1.
To eliminate the logarithm, we can rewrite the equation in exponential form:
5^1 = (2x - 1)/(x - 2).
Simplifying, we have:
5 = (2x - 1)/(x - 2).
Next, we can cross-multiply to eliminate the fraction:
5(x - 2) = 2x - 1.
Expanding and simplifying, we get:
5x - 10 = 2x - 1.
Bringing like terms to one side, we have:
5x - 2x = -1 + 10,
3x = 9.
Dividing by 3, we find:
x = 3.
Therefore, the solution to the equation is x = 3.
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Examples of maximum likelihood estimators》 For data that comes from a discrete distribution, the likelihood function is the probability of the data as a function of the unknown parameter. For data that comes from a continuous distribution, the likelihood function is the probability density function evaluated at the data, as a function of the unknown parameter, and the maximum likelihood estimator (MLE) is the parameter value that maximizes the likelihood function. For both of the questions below, write down the likelihood function and find the maximum likelihood estimator, including a justification that you have found the maximum (this involves something beyond finding a place where a derivative is 0 ). (a) If X∼Bin(n,ϑ), write the likelihood function and show that the MLE for ϑ is n
X
. (b) The exponential distribution with parameter λ (denoted by Exp(λ) ) is a continuous distribution having pdf f(t)={ λe −λt
0
t>0
t≤0.
Suppose T 1
,T 2
,…,T n
are independent random variables with T i
∼Exp(λ) for all i. Defining S=T 1
+T 2
+⋯+T n
, write the likelihood function, and show that the MLE for λ is s
n
, the reciprocal of the average of the T i
's. IITo start thinking about part (a) it may help to remember the class when we were doing inference about ϑ in a poll of size n=100 with the observed data X=56. For that example we calculated and plotted the likelihoods for ϑ=0,.001,.002,…,.998,.999,1, and it looked like the value that gave the highest likelihood was 0.56. Well, 0.56= 100
56
= n
x
in that example. Here we are thinking of the likelihood as a function of the continuous variable ϑ over the interval [0,1] and showing mathematically that ϑ
^
= n
X
maximizes the likelihood. So start by writing down the likelihood function, that is, writing the binomial probability for getting X successes in n independent trials each having success probability ϑ. Think of this as a function of ϑ (in any given example, n and X will be fixed numbers, like 100 and 56 ), and use calculus to find the ϑ
^
that maximizes this function. You should get the answer ϑ
^
= n
X
. Just as a hint about doing the maximization, you could maximize the likelihood itself, or equivalently you could maximize the log likelihood (which you may find slightly simpler).]
(a) The maximum likelihood estimator for ϑ is ϑ^ = x/n, which is the ratio of the number of successes (x) to the sample size (n).
(b) The maximum likelihood estimator for λ is λ^ = 1 / (T1 + T2 + ... + Tn), which is the reciprocal of the average of the observed values T1, T2, ..., Tn.
The maximum likelihood estimator (MLE) is a method for estimating the parameters of a statistical model based on maximizing the likelihood function or the log-likelihood function. It is a widely used approach in statistical inference.
(a) If X follows a binomial distribution with parameters n and ϑ, the likelihood function is given by:
L(ϑ) = P(X = x | ϑ) = C(n, x) * ϑ^x * (1 - ϑ)^(n - x)
To find the maximum likelihood estimator (MLE) for ϑ, we need to maximize the likelihood function with respect to ϑ. Taking the logarithm of the likelihood function (log-likelihood) can simplify the maximization process without changing the location of the maximum. Therefore, we consider the log-likelihood function:
ln(L(ϑ)) = ln(C(n, x)) + x * ln(ϑ) + (n - x) * ln(1 - ϑ)
To find the maximum, we differentiate the log-likelihood function with respect to ϑ and set it equal to 0:
d/dϑ [ln(L(ϑ))] = (x / ϑ) - ((n - x) / (1 - ϑ)) = 0
Simplifying this equation, we have:
(x / ϑ) = ((n - x) / (1 - ϑ))
Cross-multiplying, we get:
x - ϑx = ϑn - ϑx
Simplifying further:
x = ϑn
(b) Given that T1, T2, ..., Tn are independent random variables following an exponential distribution with parameter λ, the likelihood function can be written as:
L(λ) = f(T1) * f(T2) * ... * f(Tn) = λ^n * e^(-λ * (T1 + T2 + ... + Tn))
Taking the logarithm of the likelihood function (log-likelihood), we have:
ln(L(λ)) = n * ln(λ) - λ * (T1 + T2 + ... + Tn)
To find the maximum likelihood estimator (MLE) for λ, we differentiate the log-likelihood function with respect to λ and set it equal to 0:
d/dλ [ln(L(λ))] = (n / λ) - (T1 + T2 + ... + Tn) = 0
Simplifying this equation, we get:
n = λ * (T1 + T2 + ... + Tn)
Dividing both sides by (T1 + T2 + ... + Tn), we have:
λ^ = n / (T1 + T2 + ... + Tn)
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9 years ago the Queen bought a property in Queens for $28,386, today the property is worth $66,418. Estimate the average annual rate of growth over the years. The geometric sequence, or compound interest model should be used here, and we assume the growth was assessed annually.
Enter answer as a percent rounded to a whole number. For example, if the answer is 25.8%, enter 26.
The property purchased by the Queen in Queens has experienced an average annual growth rate of around 7% over the past 9 years, according to the compound interest model. This indicates a steady increase in the property's value over time.
The average annual rate of growth for the property purchased by the Queen in Queens over the past 9 years is approximately 7%. This estimation is based on the compound interest model or geometric sequence, assuming annual growth assessments.
To calculate the average annual rate of growth, we can use the formula for compound interest:
Future Value = Present Value * (1 + r)^n
In this case, the present value (P) is $28,386, the future value (F) is $66,418, and the number of years (n) is 9. We need to solve for the annual growth rate (r). Rearranging the formula, we have:
r = (F / P)^(1/n) - 1
Plugging in the values, we get:
r = ($66,418 / $28,386)^(1/9) - 1 ≈ 0.068
Converting this decimal to a percentage, we find that the average annual rate of growth is approximately 6.8%. Rounded to the nearest whole number, the answer is 7%.
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Consider the polynomial p(x)=∑j=0najxj with coefficients aj=j for j=0,…,n. When n=12, what is p(12)(7), or, in other words, what is the 12-th derivative of this polynomial evaluated at x=7 ? a. 7 b. 12 c. 157480920 d. 457801920 e. 574801920 f. 1574809200 g. 4578019200 h. 5748019200
We have a12 = 12 and p^(12)(7) = 12!a12 = 12! * 12 = 5748019200. Therefore, the answer is (h) 5748019200.
The 12-th derivative of p(x) is obtained by applying the power rule repeatedly:
p(x) = a0 + a1x + a2x^2 + ... + anx^n
p'(x) = a1 + 2a2x + 3a3x^2 + ... + nanx^(n-1)
p''(x) = 2a2 + 6a3x + ... + n(n-1)anx^(n-2)
...
p^(12)(x) = 12!a12
Since aj = j for j=0,...,n, we have a12 = 12 and p^(12)(7) = 12!a12 = 12! * 12 = 5748019200. Therefore, the answer is (h) 5748019200.
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Find the area under the standard normal distribution curve to the right of z=0.77. Use 0 The Standard Normal Distribution Table and enter the answer to 4 decimal places. The aree to the right of the z value is Find the area under the standard normal distribution curve between z=−1.31 and z=−2.73. Use (B) The Standard Normal Distribution Table and enter the answer to 4 decimal places. The area between the two z values is Find the area under the standard normal distribution curve to the right of z=−2.22, Use 3 The 5tandard Normal Distribution Table and enter the answer to 4 decimal places. The area to the right of the z value is
Area under the standard normal distribution curve is as follows:
to the right of z = 0.77 = 0.2206
Between z = −1.31 and z = −2.73 = 0.0921
to the right of z = −2.22 = 0.9861
The area under the standard normal distribution curve: To the right of z = 0.77, using the standard normal distribution table: According to the standard normal distribution table, the area to the left of z = 0.77 is 0.7794.
The total area under the curve is 1. Therefore, the area to the right of z = 0.77 can be found by subtracting 0.7794 from 1, which equals 0.2206.
Therefore, the area under the standard normal distribution curve to the right of z = 0.77 is 0.2206.
To the right of z = −2.22, using the standard normal distribution table:
According to the standard normal distribution table, the area to the left of z = −2.22 is 0.0139.
The total area under the curve is 1.
Therefore, the area to the right of z = −2.22 can be found by subtracting 0.0139 from 1, which equals 0.9861.
Therefore, the area under the standard normal distribution curve to the right of z = −2.22 is 0.9861.
Between z = −1.31 and z = −2.73, using the standard normal distribution table:
According to the standard normal distribution table, the area to the left of z = −1.31 is 0.0951, and the area to the left of z = −2.73 is 0.0030.
The area between these two z values can be found by subtracting the smaller area from the larger area, which equals 0.0921.
Therefore, the area under the standard normal distribution curve between z = −1.31 and z = −2.73 is 0.0921.
Area under the standard normal distribution curve:
To the right of z = 0.77 = 0.2206
Between z = −1.31 and z = −2.73 = 0.0921
To the right of z = −2.22 = 0.9861
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Compute The Average Rate Of Change F(X)=1/x On The Interval [4,14]. Average Rate Of Change =
The average rate of change of the function f(x) = 1/x on the interval [4, 14] is -1/560.
The function f(x) = 1/x on the interval [4, 14] is used to compute the average rate of change. Let's find the average rate of change of the function.Step 1: The average rate of change formula is given by;AROC = (f(b) - f(a)) / (b - a)Where,f(b) is the value of the function at upper limit 'b',f(a) is the value of the function at lower limit 'a',b-a is the change in x (or length of the interval)[4, 14].Step 2: Determine the value of f(4) and f(14)f(4) = 1/4f(14) = 1/14Step 3: Determine the average rate of change using the above formulaAROC = (f(b) - f(a)) / (b - a)= (1/14 - 1/4) / (14 - 4)= (-1/56) / 10= -1/560
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What list all of the y-intercepts of the graphed functions?
The coordinate of the y-intercept of the given quadratic graph is: (0, -3)
What is the coordinate of the y-intercept?The general form of the equation of a line in slope intercept form is:
y = mx + c
where:
m is slope
c is y-intercept
The general form of quadratic equations is expressed as:
y = ax² + bx + c
Now, from the term y-intercept, we know that it is the point where the graph crosses the y-axis and as such, we have the coordinate from the graph as:
(0, -3)
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if jamie's quarterly interest payments are $150 on a $12,000 loan, then what is her annual interest rate?
Jamie's annual interest rate is 5%.
To find Jamie's annual interest rate, we need to consider the relationship between the quarterly interest payments and the loan amount. Let's break it down step by step:
1. We know that Jamie's quarterly interest payments are $150. Since there are four quarters in a year, the total annual interest payments can be calculated by multiplying the quarterly payments by four: $150 * 4 = $600.
2. Now, let's determine the interest rate. We have the annual interest payment, but we need to express it as a percentage of the loan amount. The formula to calculate interest rate is (Interest Payment / Loan Amount) 100.
3. Substituting the values into the formula, we have ($600 / $12,000)
100 = 0.05 100 = 5%.
Therefore, Jamie's annual interest rate is 5%.
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The company i conidering adding a popicle machine to their plant. The machine will cot $1800 and they can
ell each popicle for $1. 25
The company would make $700 in profit if it sold 2000 popsicles.
Given that,
Cost of the popsicle machine = $1800
Selling price per popsicle = $1.25
To determine the number of popsicles you need to sell to cover the cost of the machine,
Divide the cost of the machine by the selling price per popsicle:
$1800 / $1.25 = 1440 popsicles
The company needs to sell at least 1440 popsicles to break even and cover the machine's cost.
To determine profitability, Assume the company sells 2000 popsicles. Calculate the revenue:
Revenue = Number of popsicles sold x Selling price per popsicle Revenue = 2000 x $1.25
= $2500
To calculate the profit, Subtract the cost of the machine from the revenue:
Profit = Revenue - Cost of machine
Profit = $2500 - $1800
= $700
Therefore, if the company sells 2000 popsicles, the profit would be $700.
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A basketball player makes 80 out of 120 free throws. We would estimate the probability that the player nukes the next free throw to be
Based on the observed success rate of the basketball player, we estimate the probability of making the next free throw to be approximately 0.667 or 2/3. To estimate the probability that the basketball player makes the next free throw, we can use the observed success rate.
The player made 80 out of 120 free throws, which means the success rate is 80/120 = 2/3. This indicates that, on average, the player makes 2 out of every 3 free throws.
Therefore, we would estimate the probability that the player makes the next free throw to be 2/3 or approximately 0.667.
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The price of a new car is $42 860. The expected value of the car after its eleven -year useful life is $1 500. Predict what would be the price of the car after 4 years.
The predicted price of the car after 4 years is $27,820.
To predict the price of the car after 4 years, we can assume that the car depreciates in a linear manner over its useful life.
The car's initial price is $42,860, and the expected value after 11 years is $1,500. Therefore, the car depreciates by $42,860 - $1,500 = $41,360 over 11 years.
To find the annual depreciation rate, we divide the total depreciation by the number of years:
Annual depreciation rate = Total depreciation / Number of years
= $41,360 / 11
= $3,760 per year
Now, to predict the price of the car after 4 years, we multiply the annual depreciation rate by the number of years:
Depreciation after 4 years = Annual depreciation rate * Number of years
= $3,760 * 4
= $15,040
Finally, we subtract the depreciation after 4 years from the initial price to find the predicted price:
Predicted price after 4 years = Initial price - Depreciation after 4 years
= $42,860 - $15,040
= $27,820
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Define the arrays presented in points (a) to (c) in the comment mention the fype of the aray (eg a vectoenD matrix, a column wector, a num mattix) a) a=[12245] b) b=⎣⎡12240⎦⎤=⎣⎡111222223444555⎦⎤
The array "b" is a matrix. It is represented as multiple rows and columns of numbers.
(a) The array a=[1 2 2 4 5] can be classified as a row vector.
(b) The array b=⎣⎡12240⎦⎤=⎣⎡111 222 223 444 555⎦⎤ is a matrix.
In array b, we have 5 rows and 1 column, with each element representing a separate entry in the matrix.
Let's go through the arrays presented in points (a) to (c) and identify the type of array:
a) a=[1 2 2 4 5] The array "a" is a row vector.
It is represented as a single row of numbers.
b) b=⎣⎡12240⎦⎤=⎣⎡111222223444555⎦⎤
The array "b" is a matrix. It is represented as multiple rows and columns of numbers.
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A component has a 1 in 25 chance of failing. Five components are chosen from a large batch so that the probability of failure remains constant. Probability of fewer than 3 component failing is: 0.000012 0.000088 0.999398 0.000602 Suppose the heights of female university students follow a normal distribution with a mean of 165 cm and a standard deviation of 6 cm, then 95% of female university students will have a height no more than: 151.84 cm 155.13 cm 178.16 cm 174.87 cm
approximately 95% of female university students will have a height no more than 174.87 cm (rounded to two decimal places).
To determine the height at which 95% of female university students will have a height no more than, we can use the properties of the normal distribution and the concept of z-scores.
In a normal distribution, approximately 95% of the data falls within 1.96 standard deviations from the mean (assuming a symmetric distribution). This is often referred to as the 95% confidence interval.
To calculate the specific height, we need to find the value that corresponds to the z-score of 1.96, given the mean and standard deviation of the distribution.
The formula to calculate the specific value (height) is:
Specific value = Mean + (Z-score * Standard Deviation)
In this case:
Mean = 165 cm
Standard Deviation = 6 cm
Z-score = 1.96
Plugging in these values, we get:
Specific value = 165 + (1.96 * 6)
Specific value ≈ 165 + 11.76
Specific value ≈ 176.76 cm
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Write an equation of the line satisfying the given conditions. Write the answer in slope -intercept form. The line contains the point (-6,19) and is parallel to a line with a slope of -(5)/(2).
The equation of the line in slope-intercept form is y = -5/2x + 4.
The line contains the point (-6, 19).And, it is parallel to a line with a slope of -5/2.
The slope-intercept form of a linear equation is y = mx + b where 'm' is the slope of the line and 'b' is the y-intercept of the line. Slope of two parallel lines is the same.
We have the slope of the given line which is -5/2 and we know that the line we want to find is parallel to this line.
So, the slope of the line which we want to find is also -5/2.
Therefore, the equation of the line passing through the point (-6, 19) with a slope of -5/2 is:
y = mx + b [Slope-Intercept Form]
y = -5/2 * x + b [Substitute 'm' = -5/2]
Now, we have to find the value of 'b'.
We know that the point (-6, 19) lies on the line.
So, substituting this point in the equation of the line:
y = -5/2 * x + b19 = -5/2 * (-6) + b [Substitute x = -6 and y = 19]
19 = 15 + b[Calculate]
b = 19 - 15 [Transposing -15 to the R.H.S]
b = 4
Now, we know the value of 'm' and 'b'.Therefore, the equation of the line passing through the point (-6, 19) with a slope of -5/2 is:y = -5/2 * x + 4 [Slope-Intercept Form].
Hence, the required equation of the line in slope-intercept form is y = -5/2x + 4.
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Which of these is a test of homogeneity of variance?
Box's M
Spearman's test
Welch's T
Tukey's test
The test of homogeneity of variance is Levene's test or Bartlett's test.
Levene's test and Bartlett's test are commonly used to assess whether the variances of multiple groups or samples are equal. These tests evaluate the null hypothesis that the variances are equal across groups.
Levene's test is less sensitive to departures from normality compared to Bartlett's test, and it is often used when the data deviates from a normal distribution. On the other hand, Bartlett's test assumes that the data is normally distributed.
In summary, Levene's test or Bartlett's test are the appropriate tests to evaluate the homogeneity of variance assumption.
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We want to conduct a hypothesis test for significance of each independent variable in the regression equation shown below. We conduct the test at the 0.01 significance level using a random sample of 22 items from the population. The critical values of the test statistic are plus and minus _______ Leave 3 decimal places in your answer
Y' = 1700 + 14.2X1 + 0.86X2 − 23X3
The critical values of the test statistic are plus and minus 2.878.
The hypothesis test for the significance of each independent variable in the regression equation is conducted below;Y' = 1700 + 14.2X1 + 0.86X2 − 23X3
The regression equation can be rewritten as Y' = b0 + b1X1 + b2X2 + b3X3
To carry out the hypothesis test for the significance of each independent variable in the regression equation, we need to use the t-test at a 0.01 significance level.
The t-test is conducted for each independent variable, and the null hypothesis (H0) and alternative hypothesis (Ha) are given below.H0: βi = 0Ha: βi ≠ 0Where βi represents the coefficient of the independent variable being tested.
The test statistic is calculated using the equation below.t = (bi - 0) / SE(bi)
Where bi represents the sample estimate of βi, and SE(bi) represents the standard error of bi.
The critical values of the test statistic are found using the t-distribution with 19 degrees of freedom (df = n - k - 1 = 22 - 3 - 1 = 18) and a significance level of 0.01.
Since this is a two-tailed test, we will look up the critical values for the 0.005 level of significance in the t-distribution table with 18 degrees of freedom.
The critical values are ±2.878 (rounded to three decimal places).Therefore, the critical values of the test statistic are plus and minus 2.878.
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Two Angles are Complementary when they add up to 90 degrees. The difference of two complementary angles is 26 degrees. Find the measures of the angles. The measure of the largest of the two angles is degrees. The measure of the smallest of the two angles is degrees.
Sorry for bad handwriting
if i was helpful Brainliests my answer ^_^
Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result. (Round your answer to three decimal places.) \[ y=\frac{x^{2}+3}{x}, x=1, x=6, y=0 \]
Therefore, the area of the region bounded by the graphs of the equations is 0.
The area of the region bounded by the graphs of the equations is 83.243 square units.
Let's start off by plotting the given equations on a graph:
plot{y=(x^2+3)/x, x=1, x=6, y=0}
As we can see from the graph, the region bounded by the curves is a shape that resembles a triangle, with an extra rectangular region added at the bottom to complete the figure. We can break up the figure into two smaller regions, one triangular and the other rectangular.
Let's calculate their areas separately:
Area of the triangular regionTo find the area of the triangular region, we need to find the base and height of the triangle. The base is the horizontal distance between x = 1 and x = 6, which is 6 - 1 = 5 units.
The height is the vertical distance from the x-axis to the curve
y = (x^2 + 3)/x.
To find the height, we need to find the y-intercept of the curve, which is the value of y when x = 0. Substituting x = 0 in the equation gives:
y = (0^2 + 3)/0 = undefined
This means that the curve does not intersect the y-axis, so the height of the triangle is 0.
Therefore, the area of the triangular region is:
0.5 * base * height = 0.5 * 5 * 0 = 0 square units
Area of the rectangular regionTo find the area of the rectangular region, we need to find its width and height. The width is the horizontal distance between x = 1 and x = 6, which is 6 - 1 = 5 units.
The height is the vertical distance between y = 0 and the curve y = (x^2 + 3)/x.
To find the height, we need to find the x-intercepts of the curve, which are the values of x that make y = 0. Setting y = 0 in the equation gives:
0 = (x^2 + 3)/x
Multiplying both sides by x gives:
x^2 + 3 = 0
This equation has no real solutions, so the curve does not intersect the x-axis.
Therefore, the height of the rectangle is 0. Therefore, the area of the rectangular region is:
width * height = 5 * 0 = 0 square units
Total area The total area of the region bounded by the curves is the sum of the areas of the triangular and rectangular regions:
0 + 0 = 0
Therefore, the area of the region bounded by the graphs of the equations is 0.
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Prove the remaining parts of Theorem 2.6 (Parts i-iii were shown in class). Let a,b, and c be real numbers, use the axioms of the real numbers and any theorems proved in class to show that: 1 (iv) (−a)(−b)=ab (v) ac=bc, with c=0, implies a=b
(vi) ab=0 implies either a=0 or b=0 (or both)
By proving (iv), (v), and (vi) using the axioms of real numbers and the theorems proved in class, we have completed the proof of Theorem 2.6.
To prove the remaining parts of Theorem 2.6, we'll use the axioms of real numbers and the theorems proved in class:
(iv) To prove (−a)(−b) = ab:
Starting with the left side:
(−a)(−b) = (−1)(a)(−1)(b) [Using the distributive property]
= (−1)(−1)(ab) [Using the associative property]
= 1(ab) [Since (−1)(−1) = 1]
= ab [Using the identity property of multiplication]
Therefore, (−a)(−b) = ab.
(v) To prove ac = bc, with c = 0, implies a = b:
Starting with the equation ac = bc:
ac - bc = 0 [Subtracting bc from both sides]
c(a - b) = 0 [Using the distributive property]
Since c = 0, we have:
0(a - b) = 0 [Multiplying both sides by 0]
0 = 0
This equation is always true, regardless of the values of a and b. Therefore, ac = bc, with c = 0, implies a = b.
(vi) To prove ab = 0 implies either a = 0 or b = 0 (or both):
We'll prove this by contradiction. Assume ab = 0 and both a ≠ 0 and b ≠ 0.
If a ≠ 0, then we can divide both sides of the equation ab = 0 by a, yielding:
b = 0
However, this contradicts our assumption that b ≠ 0. Therefore, our assumption that both a ≠ 0 and b ≠ 0 must be false.
Similarly, if b ≠ 0, we can divide both sides of the equation ab = 0 by b, yielding:
a = 0
Again, this contradicts our assumption that a ≠ 0. Therefore, our assumption that both a ≠ 0 and b ≠ 0 must be false.
Hence, if ab = 0, it implies either a = 0 or b = 0 (or both).
By proving (iv), (v), and (vi) using the axioms of real numbers and the theorems proved in class, we have completed the proof of Theorem 2.6.
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Set up (but do not evaluate) an integral that represents the area of the region that lies inside the first curve and outside the second curve. r=7cos(θ),r=3+cos(θ)
We can now set up the integral that represents the area of the region as follows:
∫_(5π/3)^(2π) ½ (7 cosθ)² dθ - ∫_(5π/3)^(2π) ½ (3 + cosθ)² dθ
The integral that represents the area of the region that lies inside the first curve and outside the second curve given the polar curves: r = 7 cos(θ) and r = 3 + cos(θ) is calculated as follows:
To obtain the area that lies inside the first curve and outside the second curve, we will first identify the points of intersection between the two curves. To do that, we will set
r = 7 cos(θ) equal to r = 3 + cos(θ)7 cos(θ) = 3 + cos(θ)6 cos(θ) = 3cos(θ)cos(θ) = 1/2θ = ±π/3, θ = ±5π/3
We can now set up the integral that represents the area of the region as follows:
∫_(5π/3)^(2π) ½ (7 cosθ)² dθ - ∫_(5π/3)^(2π) ½ (3 + cosθ)² dθ
Note that we took the upper limits of integration to be 2π, which is the full range of the parameter θ. This is because we want to integrate over the entire region of interest, which lies between the points of intersection.
However, we subtracted the integral of the second curve from the integral of the first curve so as to ensure that we only obtain the area between the curves and not the area outside the first curve.
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The endpoints of a diameter of a circle are (3,-7) and (-1,5). Find the center and the radius of the circle and then write the equation of the circle in standard form.
If the two endpoints of the diameter of a circle as (3, -7) and (-1, 5), then the center of the circle is (1, -1), radius of the circle is 2√10 and the equation of the circle in standard form is (x – 1)² + (y + 1)² = 40.
To find the center, radius and the equation of the circle, follow these steps:
The midpoint of the diameter is the center of the circle. So, The center is calculated as follows: Center is [(-1+3)/2, (5-7)/2] = (1, -1)Therefore, the center of the circle is (1, -1).The radius of the circle is half the length of the diameter. We can use the distance formula to find the length of the diameter. Distance between (3, -7) and (-1, 5) is calculated as follows: [tex]d = (\sqrt{(3-(-1))^2 + (-7-5)^2}) = (\sqrt{(4)^2 + (-12)^2}) = (\sqrt{(16 + 144)})= (\sqrt{160})[/tex] Therefore, d=4√10. Since the radius is half the length of the diameter, radius= 2√10.The equation of a circle in standard form is (x – h)² + (y – k)² = r², where (h, k) is the center of the circle, and r is the radius of the circle. Substituting the values in the equation of the circle, we get the equation as (x – 1)² + (y + 1)² = 40.Learn more about circle:
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, SEgMENTS AND ANGLES Table for a linear equation Fill in the table using this function rule. y=-3x+4
The table for the linear equation y = -3x + 4 is as follows:
x y
-2 10
-1 7
0 4
1 1
2 -2
To find the corresponding values for y, we substitute each x-value into the equation and evaluate the expression. For example, when x = -2, we have:
y = -3(-2) + 4
y = 6 + 4
y = 10
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The weight of an energy bar is approximately normally distributed with a mean of 42.40 grams with a standard deviation of 0.035 gram. Complete parts (a) through (e) below.
If a sample of 4 energy bars is selected, what is the probability that the sample mean weight is less than 42.375grams?
To find the probability that the sample mean weight is less than 42.375 grams, we can use the Central Limit Theorem. The Central Limit Theorem states that for a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.
(a) Find the standard deviation of the sample mean: The standard deviation of the sample mean (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard deviation of the sample mean is 0.035 grams divided by the square root of 4, which is 0.035/√4 = 0.0175 grams.
(b) Convert the given value to a z-score: To calculate the z-score, we use the formula z = (x - μ) / σ, where x is the value of interest, μ is the population mean, and σ is the standard deviation of the sample mean. In this case, x is 42.375 grams, μ is 42.40 grams, and σ is 0.0175 grams. Plugging in the values, we get z = (42.375 - 42.40) / 0.0175 = -1.43.
(c) Find the probability associated with the z-score: We can use a standard normal distribution table or a calculator to find the probability associated with the z-score of -1.43. Looking up the z-score in the table, we find that the probability is approximately 0.0764.
(d) Interpret the probability: The probability that the sample mean weight is less than 42.375 grams is approximately 0.0764, or 7.64%.
(e) There is a 7.64% probability that the sample mean weight of 4 energy bars is less than 42.375 grams.
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According to a recent survey. T3Yh of all tamilies in Canada participatod in a Hviloween party. 14 families are seiected at random. What is the probabity that wix tamilies participated in a Halloween paty? (Round the resut to five decimal places if needed)
The probability that six families participated in a Halloween party is 0.16859
As per the given statement, "T3Yh of all families in Canada participated in a Halloween party."This implies that the probability of families participating in a Halloween party is 30%.
Now, if we select 14 families randomly, the probability of selecting 6 families from the selected 14 families is determined by the probability mass function as follows:
`P(x) = (14Cx) * 0.3^x * (1 - 0.3)^(14 - x)`
where P(x) represents the probability of selecting x families that participated in a Halloween party.
Here, x = 6
Thus, `P(6) = (14C6) * 0.3^6 * (1 - 0.3)^(14 - 6)``
P(6) = 0.16859`
Hence, the probability that six families participated in a Halloween party is 0.16859.
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Find An Equation Of The Line Tangent To The Graph Of F(X)=X3/xAt (8,3/8). The Equation Of The Tangent Line Is Y=
the function, f(x) = x³/x, the first step is to find its derivative f′(x) which will help in finding the slope of the tangent at point (8,3/8).
Hence, f′(x) can be found as follows; f(x) = x³/x
ƒ(x) = x²
ƒ′(x) = 2x
Taking the point given, (8,3/8), and substituting it in the function to get the slope of the tangent;
ƒ′(8) = 2(8)
= 16
At point (8,3/8), the slope of the tangent is 16.Using the point-slope form of a linear equation, y - y₁ = m(x - x₁)
We know the point (x₁,y₁) = (8,3/8)
and the slope m = 16
Substituting into the equation, we get y - 3/8 = 16(x - 8)
Multiplying through by the common denominator of 8,y - 3 = 16x - 128
Rearranging the equation, we get y = 16x - 125
The equation of the tangent line is y = 16x - 125.
The equation of the tangent line is y = 16x - 125. Given the function, f(x) = x³/x, the first step is to find its derivative f′(x) which will help in finding the slope of the tangent at point (8,3/8).
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Compute the derivative of the following function.
h(x)=x+5 2 /7x² e^x
The given function is h(x) = x+5(2/7x²e^x).To compute the derivative of the given function, we will apply the product rule of differentiation.
The formula for the product rule of differentiation is given below. If f and g are two functions of x, then the product of these functions can be differentiated as shown below. d/dx [f(x)g(x)] = f(x)g'(x) + g(x)f'(x)
Using this formula for the given function, we have: h(x) = x+5(2/7x²e^x)\
h'(x) = [1.2/7x²e^x] + [x+5](2e^x/7x^3)
The derivative of the given function is h'(x) = [1.2/7x²e^x] + [x+5](2e^x/7x^3).
Therefore, the answer is: h'(x) = [1.2/7x²e^x] + [x+5](2e^x/7x^3).
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Let p ( x ) be a polynomial of degree n , that is, p(x) = Pn i=0 aix i 1. Describe a simple O ( n 2 ) time algorithm for computing p ( x ) . 2. Describe an O ( n log n ) time algorithm for computing p ( x ) based upon a more efficient calculation of x i 3. Now consider a rewriting of p(x)asp(x) = a0 + x(a1 + x(a2 + x(a3 + .. + x(an − 1 + x.an))) which is known as Horner’s method . Using the big-Oh notation, characterize the number of arithmetic operations this method executes.
The number of arithmetic operations executed by Horner's method can be characterized as O(n) since each coefficient ai is multiplied by x and added to the intermediate result only once. There are a total of n coefficients in the polynomial, so the number of arithmetic operations is proportional to n, resulting in O(n) complexity.
To compute the polynomial p(x) = Σ(ai * xi), a simple O(n^2) time algorithm can be used. The algorithm can be outlined as follows:
sql
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Input: Polynomial coefficients a0, a1, ..., an and value of x
Output: Value of p(x)
1. Initialize result = 0
2. For i from n to 0:
3. result = result * x + ai
4. Return result
This algorithm iterates through the coefficients of the polynomial in decreasing order, multiplying the current result by x and adding the next coefficient ai. The time complexity of this algorithm is O(n^2) because there are n iterations, and each iteration involves a multiplication and addition operation.
To compute the polynomial p(x) in O(n log n) time, a more efficient calculation of xi can be used. The algorithm can be outlined as follows:
markdown
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Input: Polynomial coefficients a0, a1, ..., an and value of x
Output: Value of p(x)
1. Initialize result = 0
2. Initialize power = 1
3. For i from 0 to n:
4. result = result + ai * power
5. power = power * x
6. Return result
This algorithm calculates xi efficiently by repeatedly squaring the current power of x. It iterates through the coefficients of the polynomial in increasing order, multiplying each coefficient ai by the corresponding power of x and adding it to the result. The time complexity of this algorithm is O(n log n) because there are n iterations, and in each iteration, the power of x is updated by squaring, which can be done in logarithmic time.
Horner's method is a more efficient way to compute the polynomial p(x) by rewriting it in a nested form. In Horner's method, the polynomial is expressed as p(x) = a0 + x(a1 + x(a2 + x(...(an-1 + x.an)...))). The number of arithmetic operations executed by Horner's method can be characterized as O(n) since each coefficient ai is multiplied by x and added to the intermediate result only once. There are a total of n coefficients in the polynomial, so the number of arithmetic operations is proportional to n, resulting in O(n) complexity.
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Derivatives Of Functions Given Implicitly 2. Find dy/dx in terms of x and y if (x ^4)y−x−5y−8=0. dy/dx
=
To find dy/dx in terms of x and y if (x ^4)y−x−5y−8=0, the chain rule is applied and then the derivative of y with respect to x is obtained. Then, the expression is simplified, and the final result is obtained. The result is dy/dx = [x³ - 5]/[4x³y - 1].
To find dy/dx in terms of x and y for the equation (x^4)y - x - 5y - 8 = 0, we will differentiate the equation implicitly with respect to x.
Differentiating implicitly with respect to x:
d/dx[(x^4)y] - d/dx[x] - d/dx[5y] - d/dx[8] = 0
To differentiate (x^4)y, we will use the product rule:
[(x^4)(dy/dx)] + [4x^3y] - 1 - 0 - 0 = 0
Simplifying the equation:
(x^4)(dy/dx) + 4x^3y - 1 = 0
Rearranging the terms:
(x^4)(dy/dx) = 1 - 4x^3y
Now, we can solve for dy/dx:
dy/dx = (1 - 4x^3y)/(x^4)
Therefore, dy/dx in terms of x and y for the equation (x^4)y - x - 5y - 8 = 0 is given by:
dy/dx = (1 - 4x^3y)/(x^4)
To find dy/dx in terms of x and y if (x ^4)y−x−5y−8=0, the chain rule is applied and then the derivative of y with respect to x is obtained. Then, the expression is simplified, and the final result is obtained. The result is dy/dx = [x³ - 5]/[4x³y - 1]. Given: (x^4)y - x - 5y - 8 = 0To find: dy/dx. To solve the given problem, we use the chain rule as follows: First, differentiate with respect to x on both sides of the given equation:(x^4)dy/dx + 4x³y - 1 - 1( dy/dx + 5) = 0.Rearrange the above equation and express dy/dx in terms of x and y: dy/dx = -4x³y + 1 / x^4 - 1.Using the given equation, replace (x^4)y by (x + 5y + 8), we get:(x + 5y + 8) - x - 5y - 8 = 0On solving this equation, we get:x + 5y = 0or, y = -x/5Substitute this value of y in the previously obtained equation for dy/dx, we get: dy/dx = [x³ - 5]/[4x³y - 1].Substituting the value of y, we get: dy/dx = [x³ - 5]/[-4x⁴/5 - 1]Multiplying numerator and denominator by -5, we get:dy/dx = [x³ - 5]/[4x³y - 1]Therefore, the solution is dy/dx = [x³ - 5]/[4x³y - 1].
To solve the given problem of finding dy/dx in terms of x and y if (x^4)y - x - 5y - 8 = 0, the chain rule is applied to find the derivative of y with respect to x, and then the expression is simplified. The problem can be solved by differentiating with respect to x on both sides of the given equation, and then applying the chain rule to find dy/dx. On solving, the final result is obtained as dy/dx = [x³ - 5]/[4x³y - 1].The chain rule states that if y is a function of u and u is a function of x, then the derivative of y with respect to x is given by dy/dx = dy/du * du/dx.In the given problem, (x^4)y - x - 5y - 8 = 0, differentiate the given equation with respect to x: d/dx [(x^4)y - x - 5y - 8] = 0.Using the product rule and chain rule, we get:(x^4)dy/dx + 4x³y - 1 - dy/dx - 5 = 0.Rearranging the above equation, we get:dy/dx = (4x³y - 1) / (x^4 - 1)Now, we have to express y in terms of x to obtain dy/dx in terms of x. Substituting the value of y, we get:dy/dx = [x³ - 5]/[-4x⁴/5 - 1] Multiplying numerator and denominator by -5, we get:dy/dx = [x³ - 5]/[4x³y - 1]. Therefore, the solution is dy/dx = [x³ - 5]/[4x³y - 1].
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1)A chemist determined the percentage of iron in an ore and obtained the following data. Mean 16. 30, deviation of 0. 20 and n=4
a Calculate the 90% confidence interval of the mean. From table 2. 5, t=2. 533
b calculate the 99% confidence interval
2) The normality of a solution was calculated with 4 separate titrations and the results were 0. 2049, 0. 2051, 0. 2458, 0. 2033. Calculate the mean, range, mean deviation, relative mean deviation, standard deviation, and coefficient of variation
1. a) The 90% confidence interval for the mean is [16.0467, 16.5533].
b) the 99% confidence interval for the mean is [15.8459, 16.7541].
2. Mean = 0.2148
Range = 0.0425
Mean Deviation = 0.015525
Relative Mean Deviation = 7.217%
Standard Deviation = 0.018605
Coefficient of Variation = 8.653%
1. Confidence Interval Calculation:
a) To calculate the 90% confidence interval for the mean, we will use the formula:
CI = X ± (t * (s / √n)),
where X is the sample mean, t is the critical value from the t-distribution table (with n-1 degrees of freedom), s is the sample standard deviation, and n is the sample size.
Given data:
X = 16.30 (sample mean)
s = 0.20 (sample standard deviation)
n = 4 (sample size)
t (for 90% confidence with 3 degrees of freedom) = 2.533 (from the t-distribution table)
Calculating the confidence interval:
CI = 16.30 ± (2.533 * (0.20 / √4))
= 16.30 ± (2.533 * 0.10)
= 16.30 ± 0.2533
= [16.0467, 16.5533]
Therefore, the 90% confidence interval for the mean is [16.0467, 16.5533].
b) To calculate the 99% confidence interval, we will use the same formula as above but with a different critical value from the t-distribution table.
t (for 99% confidence with 3 degrees of freedom) = 4.541 (from the t-distribution table)
2. Calculating the confidence interval:
CI = 16.30 ± (4.541 * (0.20 / √4))
= 16.30 ± (4.541 * 0.10)
= 16.30 ± 0.4541
= [15.8459, 16.7541]
Therefore, the 99% confidence interval for the mean is [15.8459, 16.7541].
Calculation of Various Statistical Measures:
Given data:
0.2049, 0.2051, 0.2458, 0.2033
a) Mean Calculation:
Mean = (0.2049 + 0.2051 + 0.2458 + 0.2033) / 4
= 0.8591 / 4
= 0.2148
b) Range Calculation:
Range = Maximum Value - Minimum Value
= 0.2458 - 0.2033
= 0.0425
c) Mean Deviation Calculation:
Mean Deviation = (|0.2049 - 0.2148| + |0.2051 - 0.2148| + |0.2458 - 0.2148| + |0.2033 - 0.2148|) / 4
= 0.0099 + 0.0097 + 0.031 + 0.0115 / 4
= 0.0621 / 4
= 0.015525
d) Relative Mean Deviation Calculation:
Relative Mean Deviation = (Mean Deviation / Mean) * 100
= (0.015525 / 0.2148) * 100
= 7.217%
e) Standard Deviation Calculation:
Standard Deviation = √((0.2049 - 0.2148)^2 + (0.2051 - 0.2148)^2 + (0.2458 - 0.2148)^2 + (0.2033 - 0.2148)^2) / 4
= √(0.0099^2 + 0.0097^2 + 0.031^2 + 0.0115^2) / 4
= √(0.00009801 + 0.00009409 + 0.000961 + 0.00013225) / 4
= √0.00138535 / 4
= √0.0003463375
= 0.018605
f) Coefficient of Variation Calculation:
Coefficient of Variation = (Standard Deviation / Mean) * 100
= (0.018605 / 0.2148) * 100
= 8.653%
Therefore, the calculated statistical measures are:
Mean = 0.2148
Range = 0.0425
Mean Deviation = 0.015525
Relative Mean Deviation = 7.217%
Standard Deviation = 0.018605
Coefficient of Variation = 8.653%
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