Skip to main content
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 6.2.31b
Chapter 6, Problem 6.2.31b

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (b) find the sample standard deviation
[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)
Table displaying annual earnings in dollars of 32 life insurance underwriters, with values organized in rows and columns.

Verified step by step guidance
1
Step 1: Organize the data set provided into a list of values. These represent the annual earnings of 32 intermediate-level life insurance underwriters.
Step 2: Calculate the mean (average) of the data set. The formula for the mean is \( \text{Mean} = \frac{\sum x_i}{n} \), where \( x_i \) are the individual data points and \( n \) is the total number of data points.
Step 3: Compute the deviations of each data point from the mean. For each data point \( x_i \), calculate \( x_i - \text{Mean} \).
Step 4: Square each deviation to eliminate negative values and sum all the squared deviations. This gives \( \sum (x_i - \text{Mean})^2 \).
Step 5: Divide the sum of squared deviations by \( n-1 \) (since this is a sample standard deviation) and take the square root of the result. The formula for the sample standard deviation is \( s = \sqrt{\frac{\sum (x_i - \text{Mean})^2}{n-1}} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion in a set of sample data points. It quantifies how much the individual data points deviate from the sample mean. A higher standard deviation indicates greater variability among the data points, while a lower standard deviation suggests that the data points are closer to the mean.
Recommended video:
Guided course
08:45
Calculating Standard Deviation

Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the true population parameter with a specified level of confidence, typically 95% or 99%. It provides an estimate of uncertainty around a sample statistic, allowing researchers to make inferences about the population from which the sample was drawn.
Recommended video:
06:33
Introduction to Confidence Intervals

Random Sampling

Random sampling is a technique used to select a subset of individuals from a larger population, where each individual has an equal chance of being chosen. This method helps to ensure that the sample is representative of the population, reducing bias and allowing for more accurate statistical inferences about the population based on the sample data.
Recommended video:
05:11
Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

Senate Filibuster You wish to estimate, with 99% confidence, the population proportion of U.S. adults who disapprove of the U.S Senate’s use of the filibuster. Your estimate must be accurate within 2% of the population proportion.

b. Find the minimum sample size needed, using a prior survey that found that 34% of U.S. adults disapprove of the U.S Senate’s use of the filibuster. (Source: Monmouth University)

62
views
Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (b) find the sample standard deviation. Assume the population is normally distributed.

SAT Scores The SAT scores of 12 randomly selected high school seniors

66
views
Textbook Question

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

b. Increase in the sample size

114
views
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)

78
views
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.

[IMAGE]

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

[IMAGE]

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

c. c = 0.95, xbar = 40.3, σ = 0.5, N = 300, n = 68.

68
views
Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population mean. Assume the population is normally distributed.

Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students

97
views