The lower and upper boundaries for the middle 68% of heights falls between 35 inches and 45 inches.
The empirical rule, too known as the 68-95-99.7 rule, could be a statistical rule that depicts the approximate rate of perceptions that fall within a certain number of standard deviations from the mean of a normal distribution. Particularly, the empirical rule states that for a normal distribution, around:
68% of the observations fall under one standard deviation of the mean.95% of the observations fall under two standard deviations of the mean.99.7% of the observations fall under three standard deviations of the mean.According to the Empirical rule,
Mean = 40 inches(given)
Standard deviation = 5 inches(given)
To find out the lower and upper boundaries for the middle of 68% heights-
lower boundary = mean - standard deviation
upper boundary = mean + standard deviation
So,
lower boundary = 40 - 5 = 35 inches.
upper boundary = 40 + 5 = 45 inches.
Therefore, 68% of heights fall between 35 inches and 45 inches.
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One use of regression equation is to increase the accuracy of predicting Y scores. How much
of the variability in the Y scores is predicable from the regression equation?
a. r
b. 1 - r2
c. 1 - r
d. r2
We are 95% confident that the true mean number of books people read in the past year lies between 9.52 and 11.48 books.
What is mean?
In statistics, the mean (also known as the arithmetic mean or average) is a measure of central tendency that represents the sum of a set of numbers divided by the total number of numbers in the set.
To construct a confidence interval for the mean number of books people read, we can use the formula:
CI = x ± z*(s/√n)
Where:
x is the sample mean of the number of books read in the past year
s is the sample standard deviation of the number of books read in the past year
n is the sample size
z is the z-score for the desired confidence level (95% in this case)
Plugging in the values we have:
x = 10.5
s = 16.6
n = 1018
z = 1.96 (from the standard normal distribution table for a 95% confidence level)
CI = 10.5 ± 1.96*(16.6/√1018)
CI = 10.5 ± 0.98
CI = [9.52, 11.48]
We are 95% confident that the true mean number of books people read in the past year lies between 9.52 and 11.48 books. This means that if we were to repeat the survey many times, 95% of the intervals we construct would contain the true mean number of books read.
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Convert each unit.
32
yards =
feet
1
4
foot =
inches
3
miles =
yards
Step-by-step explanation:
32 yards = 96 feet (1 yard = 3 feet, so 32 yards x 3 feet/yard = 96 feet)
14 feet = 168 inches (1 foot = 12 inches, so 14 feet x 12 inches/foot = 168 inches)
3 miles = 5,280 yards (1 mile = 1,760 yards, so 3 miles x 1,760 yards/mile = 5,280 yards)
A car travels 50 meters east in 1.0 seconds the displacement of the car at the end of this 2.0 seconds intervals is?
Answer:
100 m
Step-by-step explanation:
because if 50 m in 1.0 a than 2.0 sec it's 100
a math professor finds that when she schedules an office hour for student help, an average of 2.5 students arrive. find the probability that in a randomly selected office hour, the number of student arrivals is 4 .
The probability that 4 students arrive during a randomly selected office hour is 0.134, or about 13.4%.
To find the probability that 4 students arrive during a randomly selected office hour, we need to use the Poisson distribution formula.
The Poisson distribution is used to model the number of events that occur in a fixed interval of time or space.
The formula for the Poisson distribution is:
P(X = x) = (e^-λ * λ^x) / x!
Where X is the number of events, λ is the average number of events per interval, and e is the mathematical constant e.
In this case, λ = 2.5, since the average number of students who arrive during an office hour is 2.5. So, we can plug in λ and x = 4 into the formula:
P(X = 4) = (e^-2.5 * 2.5^4) / 4!
P(X = 4) = (0.082 * 39.0625) / 24
P(X = 4) = 0.134
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a restriction in using linear regression is that it assumes that past data and future projections fall on or near a straight lineT/F
True. Linear regression is a statistical technique that is commonly used to establish a relationship between two variables. It is based on the assumption that the relationship between the variables is linear, meaning that the data points will fall on or near a straight line.
However, this assumption can be a restriction in using linear regression, as real-world data may not always follow a linear relationship. In situations where the data is nonlinear, using linear regression can lead to inaccurate results and misleading predictions. Therefore, it is important to carefully evaluate the data and the assumptions before using linear regression, and to consider alternative statistical techniques if the data does not appear to follow a linear trend.
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the estimated annual number of smoking-attributable deaths in the united states can be broken down by specific causes, as shown: (1) (5pts) what percent of u.s. annual deaths attributable to smoking are lung cancer deaths?
According to the Centers for Disease Control and Prevention (CDC), lung cancer is the leading cause of smoking-attributable deaths in the United States.
The estimated annual number of smoking-attributable deaths in the United States is approximately 480,000. This includes deaths caused by lung cancer, as well as other smoking-related illnesses such as heart disease, stroke, and chronic obstructive pulmonary disease (COPD).
The CDC reports that about 80% of all lung cancer deaths in the United States are caused by smoking. This means that approximately 136,000 of the 170,000 annual lung cancer deaths in the United States can be attributed to smoking.
To calculate the percentage of U.S. annual deaths attributable to smoking that are lung cancer deaths, we can use the following formula:
(Lung cancer deaths attributable to smoking / Total smoking-attributable deaths) x 100
Substituting the values from above, we get:
(136,000 / 480,000) x 100 = 28.3%
Therefore, approximately 28.3% of U.S. annual deaths attributable to smoking are lung cancer deaths.
Lung cancer is a serious health problem in the United States, and smoking is the leading cause of lung cancer. According to the CDC, smoking is responsible for about 80% of all lung cancer deaths in the United States. This means that smoking is responsible for a significant proportion of all cancer deaths in the country.
In addition to lung cancer, smoking is also a major cause of other types of cancer, including throat, mouth, esophageal, pancreatic, kidney, and bladder cancer. Smoking is also a leading cause of heart disease, stroke, and COPD.
The estimated annual number of smoking-attributable deaths in the United States is approximately 480,000. This represents a staggering toll on human life, and highlights the importance of effective smoking prevention and cessation efforts.
To reduce the number of smoking-attributable deaths, it is important to implement evidence-based tobacco control policies and programs. This includes measures such as increasing the price of tobacco products, implementing smoke-free laws, and providing access to effective smoking cessation treatments. By taking action to reduce smoking rates, we can help to prevent thousands of deaths each year and improve the health and well-being of millions of Americans.
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carol successfully increases her business to 200 customers per day. however, her total cost for doing so is 50% greater than the expected $1,600. what percent greater is the actual marginal cost than the expected marginal cost, to the nearest full percent? (note: ignore the percent sign when entering your answer. for example, if your answer is 326%, enter 326.)
Answer is 50%
The expected marginal cost is $8 per customer ($1,600 total cost / 200 customers). If Carol's actual total cost for serving 200 customers is 50% greater than $1,600, her actual total cost is $2,400 (1.5 times $1,600).
To find the actual marginal cost, we divide the actual total cost by the number of customers served: $2,400 / 200 = $12 per customer.
The actual marginal cost is $4 ($12 - $8) greater than the expected marginal cost. To find what percent greater this is, we divide $4 by the expected marginal cost of $8 and multiply by 100:
$4 / $8 = 0.5
0.5 x 100 = 50
Therefore, the actual marginal cost is 50% greater than the expected marginal cost.
Answer: 50
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Solve for X and Explain
Answer:
tan(57°) = 12/x
x tan(57°) = 12
x = 12/tan(57°) = 7.793
Answer:
x ≈ 7.8
Step-by-step explanation:
using the tangent ratio in the right triangle
tan57° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{12}{x}[/tex] ( multiply both sides by x )
x × tan57° = 12 ( divide both sides by tan57° )
x = [tex]\frac{12}{tan57}[/tex] ≈ 7.8 ( to the nearest tenth )
as a nurse, part of your daily duties is to mix medications in the proper proportions for your patients. for one of your regular patients, you always mix medication a with medication b in the same proportion. last week, your patient's doctor indicated that you should mix 60 milligrams of medication a with 30 milligrams of medication b. however this week, the doctor said to only use 15 milligrams of medication b. how many milligrams of medication a should be mixed this week?
As a nurse, it is imperative to adhere to the medication dosage guidelines provided by the physician for patients. In this case, the patient's doctor has requested a change in the medication proportions to be mixed. Last week, the nurse was directed to mix 60 milligrams of medication a with 30 milligrams of medication b. However, this week, the physician has ordered the nurse to use only 15 milligrams of medication b.
To determine the appropriate dosage of medication a to be mixed this week, we must maintain the same proportion as last week but adjust for the change in the quantity of medication b.
First, we need to determine the ratio of medication a to medication b. We can do this by dividing the quantity of medication a by the quantity of medication b from last week's dosage.
60 mg / 30 mg = 2:1
This means that for every 2 milligrams of medication a, 1 milligram of medication b should be mixed.
Next, we can use this ratio to calculate the appropriate dosage of medication a for this week's prescription.
15 mg / 1 = x / 2
Where x represents the dosage of medication a.
Solving for x, we get:
x = 30 mg
Therefore, this week, the nurse should mix 30 milligrams of medication a with 15 milligrams of medication b for this patient.
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Suppose we have a loaded die that gives the outcomes 1 through 6 according to the following probability distribution.
Die outcome: 1, 2, 3, 4, 5, 6
Probability: 0.10, 0.20, 0.30, 0.2, ?, 0.1
What is the probability of rolling a 5?
Answer:
0.1
Step-by-step explanation:
the probabilities have to add up to 100%
add all probabilities and subtract from 100 to find missing value
is the sum of the integers x and y a prime number? (1)x is an even prime number. (2)y is a prime number between 10 and 20.
Based on the given information, we know that x must be 2 since it is the only even prime number. We also know that y must be either 11, 13, 17, or 19 since those are the only prime numbers between 10 and 20.
So, the sum of x and y can be 2 + 11 = 13, 2 + 13 = 15, 2 + 17 = 19, or 2 + 19 = 21.
Out of these four possible sums, only 13, 17, and 19 are prime numbers. Therefore, we can say that the sum of x and y may or may not be a prime number, depending on the specific values of x and y.
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for each this state, calculate predictions for the probability of measuring spin up and down along the x, y, and z axes, and confirm your predictions with the spins simulation. for the y and z components
To calculate the probabilities of measuring spin up and down along the y and z axes for a given quantum state:
For the x-component:
The probability of measuring spin up along the x-axis (P_x↑) is given by (|α|^2 + |β|^2)/2.
The probability of measuring spin down along the x-axis (P_x↓) is given by (|α|^2 + |β|^2)/2.
For the y-component:
The probability of measuring spin up along the y-axis (P_y↑) is given by |α|^2.
The probability of measuring spin down along the y-axis (P_y↓) is given by |β|^2.
For the z-component:
The probability of measuring spin up along the z-axis (P_z↑) is given by |α|^2.
The probability of measuring spin down along the z-axis (P_z↓) is given by |β|^2.
Please note that these calculations require knowing the coefficients α and β of the quantum state in question.
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As in Exercise 6. 72, let Y1 and Y2 be independent and uniformly distributed over the interval (0, 1). Find
a. The probability density function of U2 = max(Y1, Y2).
b. E ( U 2 ) and V (U2).
Reference
Let Y1 and Y2 be independent and uniformly distributed over the interval (0, 1). Find
a. The probability density function of U1 = min(Y1, Y2).
b. E ( U 1 ) and V (U1)
a. The probability density function of U1 = min(Y1, Y2) is 0 < u < 1
b. the value of E ( U 1 ) and V (U1) are 1/3 and 1/18 respectively.
a. To find the PDF of U1, we need to first find the cumulative distribution function (CDF) of U1. The CDF of U1 is defined as the probability that U1 is less than or equal to some value u.
P(U1 ≤ u) = P(min(Y1, Y2) ≤ u)
Since Y1 and Y2 are independent, we can write the above equation as:
P(min(Y1, Y2) ≤ u) = 1 - P(Y1 > u, Y2 > u)
Using the fact that Y1 and Y2 are uniformly distributed, we can compute the probability that they are both greater than u as:
P(Y1 > u, Y2 > u) = P(Y1 > u)P(Y2 > u) = (1 - u)(1 - u) = (1 - u)²
Therefore, the CDF of U1 is:
F(u) = 1 - (1 - u)², for 0 < u < 1.
To find the PDF of U1, we differentiate the CDF with respect to u:
f(u) = dF(u)/du = 2(1 - u), for 0 < u < 1.
Therefore, the PDF of U1 is:
f(u) = 2(1 - u), for 0 < u < 1.
b. The expected value of U1 is given by:
E(U1) = ∫ u*f(u) du, for 0 < u < 1.
Substituting the PDF of U1 into the above equation and integrating, we get:
E(U1) = ∫ u*2(1 - u) du, for 0 < u < 1.
E(U1) = [u² - (2/3)u³] from 0 to 1.
E(U1) = 1/3.
Therefore, the expected value of U1 is 1/3.
The variance of U1 is given by:
V(U1) = E(U1²) - [E(U1)]².
To find E(U1²), we use the formula:
E(U1²) = ∫ u²*f(u) du, for 0 < u < 1.
Substituting the PDF of U1 into the above equation and integrating, we get:
E(U1²) = ∫ u²*2(1 - u) du, for 0 < u < 1.
E(U1²) = [u³ - (3/4)u⁴] from 0 to 1.
E(U1²) = 1/2.
Therefore, V(U1) = E(U1²) - [E(U1)]² = (1/2) - (1/3)² = 1/18.
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Predict the shape of the distribution of the salaries of 25 chief executive officers (CEOs). A typical value is about 50 million per year, but there is an outlier at about 200 million
Choose the correct answer below
a. It should be right-skewed.
b. It should be roughly symmetric.
c. It should be bimodal
d. It should be left-skewed
The answer is a. It should be right-skewed. The distribution of the salaries of the 25 CEOs would likely be right-skewed due to the presence of the outlier at 200 million, which would cause the tail to extend towards the right (higher values).
A skewed distribution occurs when one tail is longer than the other. Skewness defines the asymmetry of a distribution. Unlike the familiar normal distribution with its bell-shaped curve, these distributions are asymmetric. The two halves of the distribution are not mirror images because the data are not distributed equally on both sides of the distribution’s peak.
Right skewed distributions occur when the long tail is on the right side of the distribution. Analysts also refer to them as positively skewed. This condition occurs because probabilities taper off more slowly for higher values. Consequently, you’ll find extreme values far from the peak on the high end more frequently than on the low.
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Suppose the results of a hypothesis test are statistically significant at the 5% level. Which of the following statements are true? (Mark all that are true.)
A.The results are definitely statistically significant at the 10% level.
B. The results are definitely statistically significant at the 1 % level.
C. The p-value is greater than 0.05.
D.The p-value is less than or equal to 0.05.
The statement that is true is D. The p-value is less than or equal to 0.05.
If the results of a hypothesis test are statistically significant at the 5% level, it means that the probability of observing the results under the null hypothesis is less than or equal to 5%. This is equivalent to saying that the p-value is less than or equal to 0.05.
Option A is not necessarily true, as the results may or may not be statistically significant at the 10% level.
Option B is not necessarily true, as the results may or may not be statistically significant at the 1% level.
Option C is not true, as we know that the p-value is less than or equal to 0.05.
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Can you find the lengths and areas and type the correct code? Please remember to type in ALL CAPS with no spaces.
Answer:
Step-by-step explanation:
LENGTHS: P, Q, R
AREAS: A, B
CODE: PLRAQB
some positive integers have exactly four positive factors. for example, 35 has only 1, 5, 7 and 35 as its factors. what is the sum of the smallest five positive integers that each have exactly four positive factors?
Answer:
The smallest five positive integers that each have exactly four factors are 6, 8, 10, 14, and 15.
6 + 8 + 10 + 14 + 15 = 53
A randomly generated list of integers from 0 to 4 is being used to simulate an event, with the number 3 representing a success. What is the estimated probability of a success?
Answer:
Step-by-step explanation:
The probability of a success is 0.2, as there are 5 possible outcomes (0 to 4), and one of them is a success (3). Therefore, the probability of a success is 1/5, or 0.2.
identify the next three terms in the geometric sequence. 8, 24, 72, 216,... 512, 1024, 4832 512, 1536, 4608 648, 1944, 3888 648, 1944, 5832
In order to determine the following three terms in the geometric series [tex]8, 24, 72, 216[/tex],..., we must first determine the common-ratio (r):
A geometric-sequence is a set of integers where each phrase following the first is obtained by multiplying the term before it by a fixed quantity known as the common- ratio (r).
Mathematical, scientific, and financial fields all use geometric sequences extensively. They can be used, for instance, to simulate population increase, radioactive isotope decay, asset depreciation, and the calculation of compound interest.
[tex]r = (24 / 8)[/tex]
[tex]r = (72 / 24)[/tex]
[tex]r = (72 / 24)[/tex]
[tex]r = (72 / 24)[/tex]
Consequently, the sequence's common ratio is [tex]3[/tex].
Following three terms are:
[tex]648 (216 * 3)[/tex]
[tex]648 (216 * 3)[/tex]
The finished sequence is thus [tex]8, 24, 72, 216, 648, 1944[/tex], and[tex]5832.[/tex]
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The inside diameter of a randomly selected piston ring is a random variable with mean value 16 cm and standard deviation 0.08 cm. (a) If X is the sample mean diameter for a random sample of n = 16 rings, where is the sampling distribution of X centered and what is the standard deviation of the X distribution?
The sampling distribution of the sample mean diameter (X) for a random sample of n = 16 rings is centered at the population mean of 16 cm. This means that on average, the sample mean diameter will be equal to the population mean. The standard deviation of the X distribution can be calculated using the formula σ/√n, where σ is the population standard deviation and n is the sample size. In this case, the population standard deviation is given as 0.08 cm and the sample size is 16. Therefore, the standard deviation of the X distribution is 0.02 cm.
The sampling distribution is important in statistical analysis as it helps to determine the probability of obtaining a particular sample mean. It also allows us to estimate the true population mean and evaluate the effectiveness of statistical inferences. In this case, we can use the sampling distribution to determine the probability of obtaining a sample mean diameter that is different from the population mean of 16 cm. With a standard deviation of 0.02 cm, we can expect that the sample means will be relatively close to the population mean, with most falling within a few standard deviations from the mean.
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An arithmetic sequence begins with −36, −30, −24, −18, −12 …
Which option below represents the formula for the sequence?
f(n) = −36 + 6(n−1)
f(n) = −36 − 6(n−1)
f(n) = −36 + 6(n+1)
f(n) = −36 − 6(n+1)
The correct formula for this arithmetic sequence is f(n) = -36 + (n-1)6. Thus, the answer is option A: f(n) = −36 + 6(n-1).
The common difference between any two consecutive terms in an arithmetic sequence is the same. In this sequence, we can observe that each term is obtained by adding 6 to the previous term. Therefore, the common difference is 6.
Now, we can use the formula for an arithmetic sequence to find the nth term of the sequence:
f(n) = a + (n-1)d
where a is the first term and d is the common difference.
In this case, the first term is -36 and the common difference is 6. Thus, the formula for the nth term of this arithmetic sequence is:
f(n) = -36 + (n-1)6
Therefore, the correct formula for this arithmetic sequence is f(n) = -36 + (n-1)6. Thus, the answer is option A: f(n) = −36 + 6(n+1).
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What is the area of this figure?
Enter your answer in the box.
units²
Answer:
36 units2
Step-by-step explanation:
To find the area of that particular figure, we need to divide it into 2 parts: part A and part B, respectively.
Since both of these parts are trapezium thus using the formula for their area is 1/2 x height x (sum of parallel sides).
We need to count the numbers on the graph to find the parallel sides.
So,
Area of A = 1/2 x h x (a+b) = 1/2 x 5 x (6+3) = 22.5 units2
Area of B = 1/2 x h x (a+b) = 1/2 x 3 x (4+5) = 13.5 units2
To find the area of the whole figure, we need to add the area of the two shapes.
So,
Area of the figure = Area of A + Area of B = 22.5 + 13.5 = 36 units2
what is the sequence of transformation matrices that map the shape on the left to the shape on the right?
To determine the sequence of transformation matrices that map the shape on the left to the shape on the right, we need to analyze the changes in the shape's position, size, and orientation.
One possible sequence of transformation matrices is:
1. Translation matrix to move the shape to the right position.
2. Scaling matrix to adjust the size of the shape.
3. Rotation matrix to rotate the shape to the desired orientation.
The specific values of each transformation matrix will depend on the precise changes between the two shapes. For example, the translation matrix will have different values depending on how far the shape needs to be moved and in which direction. Similarly, the scaling matrix will depend on the degree of change in size required, while the rotation matrix will depend on the angle of rotation needed.
By applying these three transformation matrices in the correct order, we can map the shape on the left to the shape on the right.
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Find the square root of each value. Match the tiles on the left with the appropriate tile on the right.
Sqrt 49
Sqrt 136
Sqrt 181
Sqrt 100
Sqrt -64
9
6
Not a real number
10
7
Answer:
Sqrt 49 = 7
Sqrt 136 = 11.6619037896906
Sqrt 181 = 13.45362404707371
Sqrt 100 = 10
Sqrt -64 = Not a real number (invalid input)
PLEASE HELP!!! ASAP!!
Answer: its B
Step-by-step explanation:
Are these vectors orthogonal?:v = 6i - 3jw = i+2j
The vectors v = 6i - 3j and w = i + 2j are orthogonal.
To determine if the vectors v and w are orthogonal, we need to find their dot product. If the dot product is 0, the vectors are orthogonal.
Here are the steps to find the dot product of v = 6i - 3j and w = i + 2j,
1. Identify the components of the vectors: v = (6, -3) and w = (1, 2)
2. Multiply the corresponding components of the vectors:
[tex](6 \times 1) + (-3 \times 2) = 6 - 6 [/tex]
3. Add the products: 6 - 6 = 0 Since the dot product is 0, the vectors v = 6i - 3j and w = i + 2j are orthogonal.
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Correct question is " Are these vectors orthogonal? v = 6i - 3j and w = i+2j"
What is the optimal solution for the following problem?
----------------------------------------------------
Maximize P =4x + 12y
subject to
3x + 5y ≤ 12 6x + 2y ≤ 10
and x ≥ 0, y ≥ 0.
The maximum value of P is 20.68, which occurs when x = 1.67 and y = 1.07.
The optimal solution, we need to first graph the constraints and determine the feasible region.
The first constraint is 3x + 5y ≤ 12, which represents a line with a y-intercept of 2.4 and a slope of -3/5.
The second constraint is 6x + 2y ≤ 10, which represents a line with a y-intercept of 5 and a slope of -3.
Plotting these lines on a graph, we get:
The feasible region is the shaded region that satisfies both constraints and lies in the first quadrant.
Next, we need to evaluate the objective function at each corner point of the feasible region to find the maximum value of P.
The corner points are:
(0, 2.4)
(1.67, 1.07)
(1.43, 0)
(0, 0)
Evaluating P at each of these points, we get:
(0, 2.4):
P = 9.6
(1.67, 1.07):
P = 20.68
(1.43, 0):
P = 17.72
(0, 0):
P = 0
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a graph that would indicate whether there was a negative relationship between stock and bond returns would be: group of answer choices frequency distribution pareto chart of both stock and bond returns scatter plot bar chart age of parents, college education, past vacations
A scatter plot would be the best choice to indicate whether there was a negative relationship between stock and bond returns.
What is scatter plot?A scatter plot is a type of graph used to display the relationship between two variables. It is also known as a scatter chart or scattergram. The graph consists of a series of points, each representing a single observation or measurement of two variables. The position of each point is determined by its values on the two variables. In a scatter plot, one variable is plotted on the x-axis and the other variable is plotted on the y-axis.
This type of graph would show the relationship between two variables, in this case stock and bond returns, on a single graph. It would provide a visual representation of any negative or positive correlation between the two variables.
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23) What is an employee incentive aimed at young parents?
Question 23 options:
employee college reimbursement
company provided child care
employee gym membership
discounts at expensive restaurants
One employee incentive that is aimed at young parents is company-provided child care. So, correct option is B.
Many parents struggle to balance their work responsibilities with caring for their children, especially if they cannot afford childcare services. By offering on-site child care or subsidizing the cost of childcare, employers can help ease the burden for their employees who are parents.
This incentive can also help employees who are parents to remain focused and productive while at work, knowing that their children are being well-cared for nearby.
Additionally, this benefit may make the company more attractive to potential employees who are parents, and could improve employee retention rates.
Overall, offering child care assistance as an employee incentive is a way for employers to demonstrate their support for working parents and to help them balance the demands of work and family.
So, correct option is B.
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which of the following scenarios is consistent with the expectations of the law of large numbers? (a) getting 200 threes after 600 separate rolls of a single die. (b) getting 50 twos after 600 separate rolls of a single die. (d) all of the above. (c) getting 100 sixes after 600 separate rolls of a single die. (e) none of the above.
The answer is (b) getting 50 twos after 600 separate rolls of a single die. The law of large numbers states that as the sample size increases, the sample mean approaches the population mean.
In other words, the more times you roll the die, the closer you should get to the expected value of each number on the die (which is 1/6 for a fair die). Option (a) of getting 200 threes and option (c) of getting 100 sixes are both too far away from the expected value to be consistent with the law of large numbers. Option (b) of getting 50 twos is closer to the expected value and thus consistent with the law of large numbers.
Therefore, the law of large numbers predicts that over a large number of trials, the frequency of an event should approach its probability of occurrence. In this case, the probability of rolling a two on a fair die is 1/6, so getting 50 twos after 600 rolls is consistent with the law of large numbers.
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