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.RE.4

Chapter 7, Problem 7.RE.4

Space Mountain Use the following wait times (minutes) for the Space Mountain ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B). Construct a 95% confidence interval estimate of the mean of all wait times. Write a brief statement that interprets that confidence interval.

40 35 40 40 25 80 50 30 35 40

Verified step by step guidance

Step 1: Calculate the sample mean (\( \bar{x} \)) of the given wait times. Add all the wait times together and divide by the total number of observations. The formula is \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) represents each wait time and \( n \) is the number of observations.

Step 2: Compute the sample standard deviation (\( s \)) using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \). This measures the spread of the wait times around the mean.

Step 3: Determine the standard error of the mean (\( SE \)) using the formula \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the sample standard deviation and \( n \) is the sample size.

Step 4: Identify the critical value (\( t \)) for a 95% confidence interval using a t-distribution table. The degrees of freedom (\( df \)) are \( n-1 \). For a 95% confidence level, find the corresponding \( t \)-value based on \( df \).

Step 5: Construct the confidence interval using the formula \( \text{Confidence Interval} = \bar{x} \pm t \cdot SE \). This provides the range within which the true mean wait time is likely to fall. Interpret the interval by stating that we are 95% confident the true mean wait time lies within this range.

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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, indicating how much the sample mean might vary from the actual population mean.

Sample Mean

The sample mean is the average of a set of observations or data points. It is calculated by summing all the values in the sample and dividing by the number of observations. The sample mean serves as a point estimate of the population mean and is crucial for constructing confidence intervals.

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 essential for calculating the confidence interval, as it helps determine the margin of error.

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Textbook Question

Controversial Song The song “Baby It’s Cold Outside” generated much controversy because of its lyrics and tone. CBS New York conducted a survey by asking viewers to use the Internet to respond to a question asking whether that song was really too offensive to play. Among 1043 Internet users who chose to respond, 986 said that the song was not too offensive, and 57 of the respondents said that the song was too offensive.

b. Based on the result from part (a), is it safe to say that the majority of the population does not feel that the song is too offensive.

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Textbook Question

Distributions Identify the distribution (normal, Student t, chi-square) that should be used in each of the following situations. If none of the three distributions can be used, what other method could be used?

a. In constructing a confidence interval of , you have 75 sample values and they appear to be from a population with a skewed distribution. The population standard deviation is not known.

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Textbook Question

Arm Circumferences Listed below are arm circumferences (cm) of randomly selected women (based on Data Set 1 “Body Data” from Appendix B). Also shown is the normal quantile plot of those measurements.

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

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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.

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a. Among the 514 human resource professionals who were surveyed, how many of them said that body piercings and tattoos were big personal grooming red flags?

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Textbook Question

Voting Survey In a survey of 1002 people, 70% said that they voted in a recent presidential election (based on data from ICR Research Group). Voting records show that 61% of eligible voters actually did vote.

a. Among the 1002 people surveyed, what is the actual number of people who said that they voted?

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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?

a. Assume that nothing is known about the percentage to be estimated.

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