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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.2.28a
Chapter 6, Problem 6.2.28a

Constructing a Confidence Interval In Exercises 25–28, use the data set to (a) find the sample mean. Assume the population is normally distributed.
Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students
Table displaying homework hours for 18 high school students, with values ranging from 8.8 to 15.5 hours.

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Step 1: Identify the data set provided. The weekly time spent on homework (in hours) for 18 students is given as: 12.0, 11.3, 13.5, 11.7, 12.0, 13.0, 15.5, 10.8, 12.5, 12.3, 14.0, 9.5, 8.8, 10.0, 12.8, 15.0, 11.8, 13.0.
Step 2: Calculate the sample mean. To find the sample mean, sum all the values in the data set and divide by the total number of observations (n = 18). Use the formula: μ=∑xn, where ∑x is the sum of all data points and n is the sample size.
Step 3: Add all the values in the data set. Perform the summation: ∑x=12.0+11.3+13.5+11.7+12.0+13.0+15.5+10.8+12.5+12.3+14.0+9.5+8.8+10.0+12.8+15.0+11.8+13.0.
Step 4: Divide the sum obtained in Step 3 by the sample size (n = 18). Use the formula: μ=∑x18.
Step 5: Interpret the sample mean. The sample mean represents the average weekly time spent on homework by the 18 students in the sample. This value will be used in further calculations, such as constructing a confidence interval.

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Key Concepts

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

Sample Mean

The sample mean is the average of a set of values, calculated by summing all the observations and dividing by the number of observations. In this context, it represents the average time spent on homework by the selected high school students. It is a key statistic used to estimate the population mean when the entire population cannot be measured.
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Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the population parameter (like the population mean) with a specified level of confidence, typically 95% or 99%. 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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Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this question, assuming the population is normally distributed allows for the use of specific statistical methods to calculate the confidence interval, as many statistical techniques rely on this assumption for validity.
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Related Practice
Textbook Question

Finite Population Correction Factor In Exercises 57 and 58, use the information below.

In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.

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Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is

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Use the finite population correction factor to construct each confidence interval for the population mean.

a. c = 0.99, xbar = 8.6, σ = 4.9, N = 200, n = 25.

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Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.

Drug Concentration The times (in minutes) for the drug concentration to peak when the drug epinephrine is injected into 15 randomly selected patients are listed. Use a 90% level of confidence.

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

a. Increase in the level of confidence

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Textbook Question

Juice Dispensing Machine A beverage company uses a machine to fill half-gallon bottles with fruit juice (see figure). The company wants to estimate the mean volume of water the machine is putting in the bottles within 0.25 fluid ounce.

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a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 1 fluid ounce.

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Textbook Question

Since 1935, the Gallup Organization has conducted public opinion polls in the United States and around the world. The table shows the results of Gallup’s World Affairs Poll of 2021, in which 1021 U.S. adults were polled. The remaining percentages not shown in the results are adults who were not sure.

b. What was the greatest value you obtained for p^?

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Textbook Question

Cholesterol Contents of Cheese A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.75 milligram of the population mean.

a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.10 milligrams.

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