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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.R.16
Chapter 6, Problem 6.R.16

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.
c = 0.99, s = 16.5, n = 20, xbar = 25.2

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Step 1: Identify the given values and understand the problem. Here, c = 0.99 (confidence level), s = 16.5 (sample standard deviation), n = 20 (sample size), and x̄ = 25.2 (sample mean). The goal is to calculate the margin of error and construct the confidence interval using the t-distribution.
Step 2: Determine the critical t-value (t*) for the given confidence level and degrees of freedom (df). The degrees of freedom are calculated as df = n - 1. Use a t-distribution table or statistical software to find t* corresponding to c = 0.99 and df = 19.
Step 3: Calculate the standard error of the mean (SE). The formula for SE is: SE = sn, where s is the sample standard deviation and n is the sample size.
Step 4: Compute the margin of error (ME). The formula for ME is: ME = t*SE, where t* is the critical t-value and SE is the standard error calculated in the previous step.
Step 5: Construct the confidence interval. The formula for the confidence interval is: [x̄ - ME, x̄ + ME], where x̄ is the sample mean and ME is the margin of error. Substitute the values to find the lower and upper bounds of the confidence interval.

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Key Concepts

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

Margin of Error

The margin of error quantifies the uncertainty in a sample estimate. It is calculated using the critical value from the t-distribution, the sample standard deviation, and the sample size. A higher confidence level results in a larger margin of error, indicating a wider range of plausible values for the population parameter.
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Finding the Minimum Sample Size Needed for a Confidence Interval

t-Distribution

The t-distribution is a probability distribution used when estimating population parameters when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the increased variability in smaller samples. As the sample size increases, the t-distribution approaches the normal distribution.
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Confidence Interval

A confidence interval is a range of values, derived from a sample statistic, that is likely to contain the population parameter with a specified level of confidence. It is constructed by adding and subtracting the margin of error from the sample mean. For example, a 99% confidence interval indicates that if the same sampling process were repeated, 99% of the intervals would contain the true population mean.
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Related Practice
Textbook Question

Determine the minimum sample size required to be 95% confident that the sample mean waking time is within 10 minutes of the population mean waking time. Use the population standard deviation from Exercise 1.

Textbook Question

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.98, s = 0.9, n = 12, xbar = 6.8

Textbook Question

[APPLET] The waking times (in minutes past 5:00 A.M.) of 40 people who start work at 8:00 A.M. are shown in the table at the left. Assume the population standard deviation is 45 minutes. Find (a) the point estimate of the population mean μ and (b) the margin of error for a 90% confidence interval.

Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.90, n = 16

Textbook Question

In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.

(20.75, 24.10)

Textbook Question

You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. 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 32% of U.S. adults have taken or planned to take a winter vacation in a recent year. (Source: Rasmussen Reports)