(a) Construct a 90% confidence interval for the population mean in Exercise 1. Interpret the results. (b) Does it seem likely that the population mean could be within 10% of the sample mean? Explain.

Verified step by step guidance

1. Step 1: Identify the necessary components for constructing a confidence interval. You will need the sample mean (\( \bar{x} \)), the sample standard deviation (\( s \)), the sample size (\( n \)), and the critical value (\( t^* \)) for a 90% confidence level. The critical value can be found using a t-distribution table or statistical software, based on the degrees of freedom \( df = n - 1 \).
2. Step 2: Use the formula for the confidence interval for the population mean: \( \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} \). Substitute the values for \( \bar{x} \), \( t^* \), \( s \), and \( n \) into the formula to calculate the lower and upper bounds of the confidence interval.
3. Step 3: Interpret the confidence interval. A 90% confidence interval means that if we were to take many random samples and construct confidence intervals for each, approximately 90% of those intervals would contain the true population mean. State the interval in the context of the problem.
4. Step 4: To determine if the population mean could be within 10% of the sample mean, calculate 10% of the sample mean (\( 0.1 \cdot \bar{x} \)) and check if this range falls entirely within the confidence interval. If it does, it is likely; if not, it is unlikely.
5. Step 5: Provide an explanation based on the results. If the range of 10% around the sample mean is within the confidence interval, explain why this suggests the population mean could plausibly be within that range. If not, explain why the confidence interval suggests otherwise.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, such as 90%. It is calculated using the sample mean, the standard error, and a critical value from the t-distribution or z-distribution, depending on the sample size and whether the population standard deviation is known.

Population Mean

The population mean is the average of all possible values in a population. It is a parameter that represents the central tendency of the entire group, as opposed to the sample mean, which is calculated from a subset of the population. Understanding the difference between these two means is crucial for making inferences about the population based on sample data.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It indicates how much the sample mean is expected to differ from the true population mean. In the context of confidence intervals, a smaller margin of error suggests a more precise estimate, while a larger margin of error indicates greater uncertainty about the population mean's location relative to the sample mean.