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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

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.

[APPLET] Earnings The annual earnings (in thousands of dollars) of 21 randomly selected level 1 computer hardware engineers are listed. Use a 99% level of confidence. (Adapted from Salary.com)

Textbook Question

Congress You wish to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

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.

Car Batteries The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

Textbook Question

Fast Food You wish to estimate, with 90% confidence, the population proportion of U.S. families who eat fast food at least once per week. Your estimate must be accurate within 3% of the population proportion.

b. Find the minimum sample size needed, using a prior study that found that 83% of U.S. families eat fast food at least once per week. (Source: The Barbecue Lab)

Textbook Question

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (a) find the sample mean

[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)

Textbook Question

Alcohol-Impaired Driving You wish to estimate, with 95% confidence, the population proportion of motor vehicle fatalities that were caused by alcohol-impaired driving. Your estimate must be accurate within 5% of the population proportion.

b. Find the minimum sample size needed, using a prior study that found that 28% of motor vehicle fatalities were caused by alcohol-impaired driving. (Source: National Highway Traffic Safety Administration)