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Let's analyze this problem step-by-step.

The task involves evaluating confidence intervals from random samples to see how many contain the true population mean, \(\mu = 140\). Here’s how to approach each part:

### Part (a): 80% Confidence Intervals

1. **Objective:** Determine how many of the 80% confidence intervals contain the population mean \(\mu = 140\).
2. **Method:** Look at each sample's 80% confidence interval in the table and/or graph and check if the interval includes 140.
3. **Answering**: Count the intervals that contain 140.

### Part (b): 95% Confidence Intervals

1. **Objective:** Determine how many of the 95% confidence intervals contain \(\mu = 140\).
2. **Method:** Look at each sample’s 95% confidence interval in the table and/or graph and check if the interval includes 140.
3. **Answering**: Count the intervals that contain 140.

### Part (c): Choose All True Statements

For part (c), we will examine each statement to determine if it is true based on the information provided.

### Instructions to Solve

#### Step-by-Step Analysis of Intervals

Go through the table or the graph:

1. For **80% confidence intervals**, observe the lower and upper limits in the table for each sample (S1 to S20).
2. For **95% confidence intervals**, repeat this process with the 95% limits.

Count the number of intervals that contain 140 in each case.

Would you like a detailed count of which specific intervals contain 140, or is this overview sufficient?

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### Practice Questions

1. How do you calculate a confidence interval for a given confidence level?
2. Why would a 95% confidence interval typically be wider than an 80% confidence interval?
3. How does the sample size affect the width of the confidence interval?
4. What is the interpretation of a confidence interval that does not contain the population mean?
5. How does changing the confidence level affect the probability of containing the population mean?

### Tip

Remember: a confidence interval at a higher confidence level (e.g., 95%) is wider because it aims to capture the true mean with a higher probability than a narrower interval at a lower confidence level (e.g., 80%).

Suppose we are interested in studying a population to estimate its mean. The population is normal and has a standard deviation of σ = 23. We have taken a random sample of size n = 95 from the population. This is Sample 1 in the table below. As shown in the table, the sample mean of Sample 1 is x̄ = 142.0. Also shown are the lower and upper limits of the 80% confidence interval for the population mean using this sample, as well as the lower and upper limits of the 95% confidence interval. Suppose that the true mean of the population is μ = 140. Then complete parts (a) through (c) below the table.

Learn how to evaluate 80% and 95% confidence intervals for a population mean, given a sample size of n = 95 with a true mean μ = 140. The problem explores normal distribution properties and the application of confidence interval formulas in statistical analysis.

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