SAT High School

The breaking strengths of cables produced by a certain manufacturer have a standard deviation of 92 pounds. A random sample of 90 newly manufactured cables has a mean breaking strength of 1700 pounds. Based on this sample, find a 95% confidence interval for the true mean breaking strength of all cables produced by this manufacturer. Then give its lower limit and upper limit. Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place

Answers

Answer 1

Answer:

To find the 95% confidence interval for the true mean breaking strength of all cables produced by the manufacturer, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value * Standard Error)

First, let's calculate the standard error, which is the standard deviation divided by the square root of the sample size:

Standard Error = Standard Deviation / √(Sample Size)

= 92 / √(90)

≈ 9.685

Next, we need to determine the critical value corresponding to a 95% confidence level. Since the sample size is large (n > 30), we can use the z-table. The z-value for a 95% confidence level is approximately 1.96.

Now we can calculate the confidence interval:

Confidence Interval = Sample Mean ± (Critical Value * Standard Error)

= 1700 ± (1.96 * 9.685)

Lower Limit = 1700 - (1.96 * 9.685)

≈ 1679.69

Upper Limit = 1700 + (1.96 * 9.685)

≈ 1720.31

Therefore, the 95% confidence interval for the true mean breaking strength of all cables produced by this manufacturer is approximately (1679.7, 1720.3). The lower limit is 1679.7 pounds, and the upper limit is 1720.3 pounds.

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