Question

Construct a 90​% confidence interval to estimate the population

mean when x=132 and s​ = 34 for the sample sizes below. ​
a) n=30
​b) n=50
​c) n=90

Answer 1

To construct a 90% confidence interval for the population mean with sample mean (x) of 132 and sample standard deviation (s) of 34, use the Student's t-distribution for three different sample sizes (n=30, n=50, n=90) and apply the formula CI = x ± (t* × s/√n). As the sample size increases, the confidence interval becomes narrower, while it becomes wider with a smaller sample size.

Step-by-step explanation:

When constructing a 90% confidence interval to estimate the population mean, the formula used depends on whether the population standard deviation is known or unknown. Since we are given a sample standard deviation (s) instead of the population standard deviation, we will use the Student's t-distribution.

The formula for a confidence interval is:

CI = x ± (t* × s/√n)

Where:

• x is the sample mean
• t* is the t-score from the t-distribution corresponding to the desired confidence level and degrees of freedom (n-1)
• s is the sample standard deviation
• n is the sample size

We find the t-score using the degrees of freedom (n-1) and the confidence level (90% in this case) from a t-distribution table or a statistical software.

Steps to calculate the confidence intervals for each sample size:

1. Determine the sample mean (x), which is 132.
2. Find the sample standard deviation (s), which is 34.
3. Find the t-score corresponding to a 90% confidence level for each sample size's degrees of freedom (n-1).
4. Calculate the margin of error (ME) using the formula: ME = t* × s/√n.
5. Finally, construct the confidence interval using the formula: CI = x ± ME.

As the sample size increases, the margin of error typically decreases, leading to a narrower confidence interval. Conversely, a smaller sample size usually leads to a wider confidence interval with a greater margin of error.