Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 5

Chapter 7, Problem 5

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%.

Verified step by step guidance

Step 1: Identify the critical value (tα/2). To find the critical value, use the t-distribution table or a statistical calculator. The degrees of freedom (df) are calculated as n - 1, where n is the sample size. Here, df = 36 - 1 = 35. For a 95% confidence level, find the tα/2 value corresponding to df = 35.

Step 2: Calculate the margin of error (E). The formula for the margin of error is E = tα/2 * (s / √n), where s is the sample standard deviation, n is the sample size, and tα/2 is the critical value found in Step 1. Substitute the given values: s = 695.5, n = 36, and the tα/2 value from Step 1.

Step 3: Find the confidence interval estimate of μ. The confidence interval is calculated as (x̄ - E, x̄ + E), where x̄ is the sample mean and E is the margin of error calculated in Step 2. Substitute x̄ = 3150.0 and the value of E from Step 2.

Step 4: Write the confidence interval in interval notation. Express the confidence interval as a range, such as (lower bound, upper bound), using the results from Step 3.

Step 5: Interpret the confidence interval. Write a brief statement explaining the meaning of the confidence interval. For example, 'We are 95% confident that the true mean weight of newborn girls lies within the calculated interval.'

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Key Concepts

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

Critical Value (t-distribution)

The critical value, denoted as tα/2, is a point on the t-distribution that corresponds to a specified level of confidence. It is used to determine the margin of error in confidence intervals, especially when the sample size is small or the population standard deviation is unknown. For a 95% confidence level, this value helps to establish the range within which the true population parameter is likely to fall.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It is calculated by multiplying the critical value by the standard error of the sample mean. This value indicates how much the sample mean may differ from the true population mean, providing a range around the sample mean that reflects this uncertainty.

Confidence Interval

A confidence interval is a range of values derived from sample statistics that is likely to contain the true population parameter with a specified level of confidence. It is calculated by adding and subtracting the margin of error from the sample mean. For example, a 95% confidence interval suggests that if the same sampling process were repeated multiple times, approximately 95% of the calculated intervals would contain the true mean.

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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?

Textbook Question

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.

Textbook Question

Degrees of Freedom

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

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Textbook Question

Second-Hand Smoke Refer to Data Set 15 “Passive and Active Smoke” and construct a 95% confidence interval estimates of the mean cotinine level in each of three samples: (1) people who smoke; (2) people who don’t smoke but are exposed to tobacco smoke at home or work; (3) people who don’t smoke and are not exposed to smoke. Measuring cotinine in people’s blood is the most reliable way to determine exposure to nicotine. What do the confidence intervals suggest about the effects of smoking and second-hand smoke?

Textbook Question

Heights of Presidents Refer to Data Set 22 “Presidents” in Appendix B. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a 95% confidence interval estimate of the population percentage. Based on the result, does it appear that greater height is an advantage for presidential candidates? Why or why not?

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?