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 7.9b

Chapter 7, Problem 7.9b

Alcohol in Children’s Movies Listed below is a simple random sample of times (seconds) that animated children’s movies showed the use of alcohol (based on Data Set 20 “Alcohol and Tobacco in Movies” in Appendix B).

b. Are the requirements for constructing a 95% confidence interval estimate of the population standard deviation satisfied? If so, construct that confidence interval.

Verified step by step guidance

Step 1: Verify the requirements for constructing a confidence interval for the population standard deviation. The requirements include: (a) the sample must be a simple random sample, and (b) the population must follow a normal distribution. Since the problem states that the sample is random, we need to check whether the data appears to be normally distributed. This can be done by creating a histogram or performing a normality test.

Step 2: Calculate the sample standard deviation (s) using the formula: s = sqrt((Σ(x_i - x̄)^2) / (n - 1)), where x_i represents each data point, x̄ is the sample mean, and n is the sample size. First, compute the sample mean (x̄) by summing all the data points and dividing by the sample size.

Step 3: Determine the degrees of freedom (df) for the chi-square distribution. The degrees of freedom are calculated as df = n - 1, where n is the sample size.

Step 4: Use the chi-square distribution to find the critical values for a 95% confidence interval. The critical values are obtained from a chi-square table or using statistical software, corresponding to the lower and upper tails of the distribution (α/2 and 1 - α/2, where α = 0.05).

Step 5: Construct the confidence interval for the population standard deviation using the formula: CI = [sqrt((df * s^2) / χ²_upper), sqrt((df * s^2) / χ²_lower)], where χ²_upper and χ²_lower are the critical values from the chi-square distribution, df is the degrees of freedom, and s is the sample standard deviation.

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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 with a specified level of confidence, typically 95%. It provides an estimate of uncertainty around the sample mean or standard deviation, allowing researchers to infer about the population from which the sample was drawn.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. It is crucial for constructing confidence intervals as it reflects the variability in the data.

Simple Random Sample

A simple random sample is a subset of individuals chosen from a larger set, where each individual has an equal chance of being selected. This method helps ensure that the sample is representative of the population, which is essential for making valid inferences about the population parameters, such as the mean or standard deviation.

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Related Practice

Textbook Question

7. FRESHMAN 15 Here is a sample of amounts of weight change (kg) of college students in their freshman year (from Data Set 13 “Freshman 15” in Appendix B): 11, 3, 0, –2, where –2 represents a loss of 2 kg and positive values represent weight gained. Here are ten bootstrap samples:

{11, 11, 11, 0}, {11, –2, 0, 11}, {11, –2, 3, 0}, {3, –2, 0, 11}, {0, 0, 0, 3}, {3, –2, 3, –2}, {11, 3, –2, 0}, {–2, 3, –2, 3}, {–2, 0, –2, 3}, {3, 11, 11, 11}.

b. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the standard deviation of the weight changes for the population.

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Textbook Question

Comparing Waiting Lines

The values listed below are waiting times (in minutes) of customers at the Bank of Providence, where customers may enter any one of three different lines that have formed at three teller windows. Construct a 95% confidence interval for the population standard deviation sigma.

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Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

OxyContin The drug OxyContin (oxycodone) is used to treat pain, but it is dangerous because it is addictive and can be lethal. In clinical trials, 227 subjects were treated with OxyContin and 52 of them developed nausea (based on data from Purdue Pharma L.P.).

b. Compare the result from part (a) to this 95% confidence interval for 5 subjects who developed nausea among the 45 subjects given a placebo instead of OxyContin: . What do you conclude?

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Textbook Question

Mean Pulse Rate of Males Data Set 1 “Body Data” in Appendix B includes pulse rates of 153 randomly selected adult males, and those pulse rates vary from a low of 40 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult males. Assume that we want 99% confidence that the sample mean is within 2 bpm of the population mean.

b. Assume that sigma=11.3 bpm, based on the value of s=11.3 bpm for the sample of 153 male pulse rates.

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Textbook Question

Astrology A sociologist plans to conduct a survey to estimate the percentage of adults who believe in astrology. How many people must be surveyed if we want a confidence level of 99% and a margin of error of four percentage points?

b. Use the information from a previous Harris survey in which 26% of respondents said that they believed in astrology.

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Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Tennis Challenges In a recent U. S. Open tennis tournament, women playing singles matches used challenges on 137 calls made by the line judges. Among those challenges, 33 were found to be successful with the call overturned.

b. Compare the result from part (a) to this 99% confidence interval for the percentage of successful challenges made by the men playing singles matches: . Does it appear that either gender is more successful than the other?

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