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Ch. 7 - Estimating Parameters and Determining Sample Sizes
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 7 - Estimating Parameters and Determining Sample SizesProblem 2b
Chapter 7, Problem 2b

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.
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Degrees of Freedom


b. Find the critical value ta/2 corresponding to a 95% confidence level.

Verified step by step guidance
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Step 1: Understand the problem. You are tasked with finding the critical value tα/2 corresponding to a 95% confidence level. This value is used in constructing confidence intervals when the sample size is small and the population standard deviation is unknown.
Step 2: Identify the degrees of freedom (df). The degrees of freedom for a t-distribution are calculated as df = n - 1, where n is the sample size. From the image, n = 36, so df = 36 - 1 = 35.
Step 3: Recognize the significance level (α). For a 95% confidence level, α = 1 - 0.95 = 0.05. Since the critical value tα/2 splits the remaining 5% equally in the two tails of the t-distribution, the area in each tail is α/2 = 0.05/2 = 0.025.
Step 4: Use a t-distribution table or statistical software to find the critical value tα/2. Look up the value corresponding to df = 35 and a tail probability of 0.025. Alternatively, use a calculator or software to compute this value.
Step 5: Interpret the critical value. The critical value tα/2 is the point on the t-distribution where the area to the right of it in the upper tail is 0.025. This value is used to calculate the margin of error in the confidence interval.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a certain confidence level, such as 95%, indicating the probability that the interval will capture the true parameter if the same sampling method is repeated multiple times.
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Introduction to Confidence Intervals

Critical Value (tα/2)

The critical value tα/2 is a point on the t-distribution that corresponds to the desired confidence level. For a 95% confidence level, it represents the value that separates the most extreme 2.5% of the distribution on each tail, which is used to calculate the margin of error in constructing the confidence interval.
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Critical Values: t-Distribution

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values that can vary in a statistical calculation. In the context of t-distributions, it is typically calculated as n - 1, where n is the sample size. This concept is crucial for determining the appropriate critical value from the t-distribution table.
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Critical Values: t-Distribution
Related Practice
Textbook Question

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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Textbook Question

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.

Interpreting a Confidence Interval The results in the screen display are based on a 95% confidence level. Write a statement that correctly interprets the confidence interval.

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Textbook Question

In Exercises 5–8, (a) identify the critical value ta/2 used for finding the margin of error, (b) find the margin of error, (c) find the confidence interval estimate of u, and (d) write a brief statement that interprets the confidence interval.


Birth Weights Here are summary statistics for randomly selected weights of newborn girls: n=36, x=3150.0g, s=695.5g (based on Data Set 6 “Births” in Appendix B). Use a confidence level of 95%.

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Textbook Question

Mean IQ of Data Scientists See the preceding exercise, in which we can assume that sigma=15 for the IQ scores. Data scientists are a group with IQ scores that vary less than the IQ scores of the general population. Find the sample size needed to estimate the mean IQ of data scientists, given that we want 98% confidence that the sample mean is within 2 IQ points of the population mean. Does the sample size appear to be practical?

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Textbook Question

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.

Refer to the accompanying screen display.


a. Express the confidence interval in the format that uses the “less than” symbol. Round the confidence interval limits given that the original times are all rounded to one decimal place.

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