Textbook Question
(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.
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7. Sampling Distributions & Confidence Intervals: Mean
Confidence Intervals for Population Mean
1:30 minutesProblem 6.R.11
Textbook Question
In Exercises 9–12, find the critical value tc for the level of confidence c and sample size n.
c = 0.98, n = 15
Verified step by step guidance1
Determine the degrees of freedom (df) for the t-distribution. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, df = 15 - 1.
Identify the level of confidence (c). Here, c = 0.98, which means the area in the middle of the t-distribution is 0.98, leaving 0.02 in the two tails combined.
Divide the remaining area (0.02) equally between the two tails to find the area in one tail. This is 0.02 / 2 = 0.01.
Use a t-distribution table or a statistical calculator to find the critical value (tc) that corresponds to the area in one tail (0.01) and the degrees of freedom (df = 14).
Verify the critical value (tc) by ensuring it matches the level of confidence (c = 0.98) and the degrees of freedom (df = 14).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Value
A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. In the context of confidence intervals, it represents the value that separates the confidence level from the tail probabilities. For a given confidence level, it helps determine the margin of error in estimating population parameters.
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t-Distribution
The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used when the sample size is small (typically n < 30) and the population standard deviation is unknown. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.
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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 the t-distribution, degrees of freedom are calculated as n - 1, where n is the sample size. This value is crucial for determining the appropriate critical value from the t-distribution table.
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