Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.1.29

Chapter 7, Problem 7.1.29

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?

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Step 1: Identify the data set and determine the number of presidents who were taller than their opponents. Count the total number of presidents in the sample and the number of presidents who were taller than their opponents. Let this count be denoted as x, and the total number of presidents as n.

Step 2: Calculate the sample proportion (p̂) of presidents who were taller than their opponents using the formula:

p̂ = \frac{x}{n}

. This gives the proportion of taller presidents in the sample.

Step 3: Use the sample proportion to calculate the standard error (SE) for the proportion. The formula for the standard error is:

SE = \sqrt{\frac{p̂}{(}}

Step 4: Construct the 95% confidence interval for the population proportion using the formula:

CI = p̂ ± z SE

, where z is the critical value for a 95% confidence level (approximately 1.96). Substitute the values of p̂ and SE to calculate the confidence interval.

Step 5: Interpret the confidence interval. If the interval suggests that the proportion of taller presidents is significantly greater than 50%, it may indicate that greater height is an advantage. Discuss whether the confidence interval supports this conclusion and provide reasoning based on the interval's range.

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Key Concepts

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

Proportion

In statistics, a proportion is a type of ratio that represents a part of a whole. It is calculated by dividing the number of favorable outcomes by the total number of observations. In this context, it refers to the fraction of presidents who were taller than their opponents, which helps in understanding the relationship between height and electoral success.

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, typically 95%. It provides an estimate of uncertainty around the sample proportion, allowing researchers to infer about the population from which the sample was drawn. In this case, it will help assess the population percentage of taller presidents.

Statistical Significance

Statistical significance refers to the likelihood that a relationship observed in data is not due to random chance. In the context of this question, it involves evaluating whether the proportion of taller presidents significantly influences electoral outcomes. Understanding this concept is crucial for interpreting the results of the confidence interval and determining if height is a meaningful factor in presidential elections.

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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

Finding Critical Values

In Exercises 5–8, find the critical value z=a/2 that corresponds to the given confidence level.

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Textbook Question

Seating Choice In a 3M Privacy Filters poll, respondents were asked to identify their favorite seat when they fly, and the results include these responses: window, window, other, other. Letting “window” and letting “other”, those four responses can be represented as {1, 1, 0, 0}. Here are ten bootstrap samples for those responses: [Image]

Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the proportion of respondents who indicated their favorite seat is “window.”

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%.

Textbook Question

Professor Evaluation Scores Listed below are student evaluation scores of professors from Data Set 28 “Course Evaluations” in Appendix B. Construct a 95% confidence interval estimate of for each of the two data sets. Does there appear to be a difference in variation?

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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?