In Exercises 1–4, refer to the accompanying screen display that results from a simple random sample of times (minutes) between eruptions of the Old Faithful geyser. The confidence level of 95% was used.

Degrees of Freedom

a. What is the number of degrees of freedom that should be used for finding the critical value ta/2?

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Step 1: Understand the concept of degrees of freedom (df). In the context of a t-distribution, degrees of freedom are calculated as the sample size (n) minus 1. This is because the t-distribution accounts for variability in the sample mean estimation.

Step 2: Identify the sample size (n) from the screen display. From the image provided, the sample size is given as n = 36.

Step 3: Apply the formula for degrees of freedom: df = n - 1. Substitute the value of n into the formula.

Step 4: Use the degrees of freedom (df) to find the critical value tα/2 for a 95% confidence level. This involves consulting a t-distribution table or using statistical software.

Step 5: Note that the critical value tα/2 depends on the degrees of freedom and the confidence level. For a 95% confidence level, the area in each tail of the t-distribution is α/2 = 0.025.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of estimating population parameters, df is typically calculated as the sample size minus one (n - 1). This concept is crucial for determining the appropriate critical value from the t-distribution when constructing confidence intervals or conducting hypothesis tests.

Critical Value

The critical value is a point on the scale of the test statistic that separates the region where the null hypothesis is rejected from the region where it is not rejected. For a t-distribution, the critical value is determined based on the desired confidence level and the degrees of freedom. In this case, with a 95% confidence level, the critical value will help define the margin of error for the confidence interval around the sample mean.

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the population parameter with a specified level of confidence. It is calculated using the sample mean, the critical value, and the standard error. In the provided example, the confidence interval for the mean time between eruptions of the Old Faithful geyser is given as (85.74, 91.76), indicating that we can be 95% confident that the true mean lies within this range.

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