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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 7.c.10a
Chapter 7, Problem 7.c.10a

Tour de France Listed below are the average speeds (km/h) of winners of the Tour de France men’s bicycle race. The speeds are listed in order by year, beginning with the year 2000.

a. Construct a 95% confidence interval estimate of the population mean.

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Step 1: Identify the sample data provided in the problem. These are the average speeds (in km/h) of the Tour de France winners from the year 2000 onward. Denote the sample size as n, the sample mean as \( \bar{x} \), and the sample standard deviation as \( s \).
Step 2: Determine the confidence level, which is 95% in this case. The corresponding critical value (\( t^* \)) can be found using a t-distribution table or statistical software, based on the degrees of freedom \( df = n - 1 \).
Step 3: Use the formula for the confidence interval of the population mean: \( \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( t^* \) is the critical value, \( s \) is the sample standard deviation, and \( n \) is the sample size.
Step 4: Plug the values of \( \bar{x} \), \( t^* \), \( s \), and \( n \) into the formula. Compute the margin of error \( ME = t^* \cdot \frac{s}{\sqrt{n}} \). Then calculate the lower bound as \( \bar{x} - ME \) and the upper bound as \( \bar{x} + ME \).
Step 5: Interpret the result. The 95% confidence interval provides a range of values within which the true population mean of the average speeds is likely to fall, with 95% confidence.

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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 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 true population mean.
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Introduction to Confidence Intervals

Sample Mean

The sample mean is the average of a set of observations drawn from a larger population. It serves as a point estimate of the population mean and is calculated by summing all sample values and dividing by the number of observations. The sample mean is crucial for constructing confidence intervals.
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Standard Error

The standard error measures the variability of the sample mean from the true population mean. It is calculated by dividing the sample standard deviation by the square root of the sample size. A smaller standard error indicates that the sample mean is a more accurate estimate of the population mean, which is essential for constructing a reliable confidence interval.
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Related Practice
Textbook Question

E-Cigarettes A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.7% of them being regular users of e-cigarettes. Because e-cigarette use is relatively new, there is a need to obtain today’s usage rate. How many adults must be surveyed now if we want a confidence level of 95% and a margin of error of 1.5 percentage points?


c. Does the use of the result from the 2014 survey have much of an effect on the sample size?

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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.


d. Are the survey results consistent with the actual voter turnout of 61%? Why or why not?

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Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?


c. Given that the required sample size is relatively small, could you simply survey the people that you know?

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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.


Touch Therapy When she was 9 years of age, Emily Rosa did a science fair experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily’s hand without seeing it and without touching it. Among 280 trials, the touch therapists were correct 123 times (based on data in “A Close Look at Therapeutic Touch,” Journal of the American Medical Association, Vol. 279, No. 13).



c. Using Emily’s sample results, construct a 99% confidence interval estimate of the proportion of correct responses made by touch therapists.


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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.


Job Interviews In a Harris poll of 514 human resource professionals, 45.9% said that body piercings and tattoos were big personal grooming red flags.


c. Repeat part (b) using a confidence level of 80%.


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Textbook Question

Women Who Give Birth An epidemiologist plans to conduct a survey to estimate the percentage of women who give birth. How many women must be surveyed in order to be 99% confident that the estimated percentage is in error by no more than two percentage points?



c. What is wrong with surveying randomly selected adult women?

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