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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 7.2.17
Chapter 7, Problem 7.2.17

Genes Samples of DNA are collected, and the four DNA bases of A, G, C, and T are coded as 1, 2, 3, and 4, respectively. The results are listed below. Construct a 95% confidence interval estimate of the mean. What is the practical use of the confidence interval?


2 2 1 4 3 3 3 3 4 1

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Step 1: Calculate the sample mean (\( \bar{x} \)) of the given data. Add all the values in the dataset (2, 2, 1, 4, 3, 3, 3, 3, 4, 1) and divide by the total number of observations (n = 10). The formula is \( \bar{x} = \frac{\sum x_i}{n} \).
Step 2: Calculate the sample standard deviation (s). Use the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( x_i \) represents each data point, \( \bar{x} \) is the sample mean, and \( n \) is the sample size.
Step 3: Determine the critical value (t*) for a 95% confidence level. Since the sample size is small (n = 10), use the t-distribution table with degrees of freedom \( df = n - 1 \) (in this case, \( df = 9 \)). Look up the t-value corresponding to a 95% confidence level.
Step 4: Calculate the margin of error (ME) using the formula \( ME = t^* \cdot \frac{s}{\sqrt{n}} \), where \( t^* \) is the critical value, \( s \) is the sample standard deviation, and \( n \) is the sample size.
Step 5: Construct the confidence interval. The 95% confidence interval is given by \( \bar{x} \pm ME \), where \( \bar{x} \) is the sample mean and \( ME \) is the margin of error. Interpret the interval in the context of the problem, explaining that it provides a range of plausible values for the true mean of the DNA base codes.

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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.
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Sample Mean

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

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of confidence intervals, it helps quantify the variability of the sample data, which is essential for determining the width of the confidence interval and thus the precision of the estimate.
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Related Practice
Textbook Question

Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes a sample of 106 body temperatures having a mean of and a standard deviation of 0.62F (for day 2 at 12 AM). Construct a 95% confidence interval estimate of the standard deviation of the body temperatures for the entire population.

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Textbook Question

Mercury in Sushi An FDA guideline is that the mercury in fish should be below 1 part per million (ppm). Listed below are the amounts of mercury (ppm) found in tuna sushi sampled at different stores in New York City. The study was sponsored by the New York Times, and the stores (in order) are D’Agostino, Eli’s Manhattan, Fairway, Food Emporium, Gourmet Garage, Grace’s Marketplace, and Whole Foods. Construct a 98% confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna sushi?


0.56 0.75 0.10 0.95 1.25 0.54 0.88

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Textbook Question

Sample Size for Proportion Find the sample size required to estimate the percentage of statistics students who take their statistics course online. Assume that we want 95% confidence that the proportion from the sample is within two percentage points of the true population percentage.

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Textbook Question

Constructing and Interpreting Confidence Intervals. In Exercises 13–16, use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion p; (b) identify the value of the margin of error E; (c) construct the confidence interval; (d) write a statement that correctly interprets the confidence interval.


Tennis Challenges In a recent U.S. Open tennis tournament, men playing singles matches used challenges on 240 calls made by the line judges. Among those challenges, 88 were found to be successful with the call overturned. Construct a 95% confidence interval for the proportion of successful challenges.

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Textbook Question

Confidence Levels

Given specific sample data, such as the data given in Exercise 1, which confidence interval is wider: the 95% confidence interval or the 80% confidence interval? Why is it wider?

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Textbook Question

Los Angeles Commute Time Listed below are 15 Los Angeles commute times (based on a sample from Data Set 31 “Commute Times” in Appendix B). Construct a 99% confidence interval estimate of the population mean. Is the confidence interval a good estimate of the population mean?


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