Textbook Question
Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.
c = 0.98, n = 26
94
views
Ch. 6 - Confidence Intervals
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 6, Problem 6.2.31
Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (c) construct a 98% confidence interval for the population mean.
[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)
Verified step by step guidance1
Step 1: Calculate the sample mean (x̄) by summing all the earnings data provided in the table and dividing by the total number of data points (n = 32). Use the formula: .
Step 2: Calculate the sample standard deviation (s) using the formula: , where x̄ is the sample mean and n is the sample size.
Step 3: Determine the critical value (z*) for a 98% confidence level. For a two-tailed test, look up the z-value corresponding to 98% confidence in a standard normal distribution table. This value is typically around 2.33.
Step 4: Calculate the margin of error (E) using the formula: , where z* is the critical value, s is the sample standard deviation, and n is the sample size.
Step 5: Construct the confidence interval for the population mean using the formula: , where x̄ is the sample mean and E is the margin of error.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
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 population parameter with a specified level of confidence. For example, a 98% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 98% of those intervals would contain the true population mean.
Recommended video:
06:33
Introduction to Confidence Intervals
Population Mean
The population mean is the average of a set of values for an entire population. It is a parameter that represents the central tendency of the population data. In the context of the question, it refers to the average annual earnings of all life insurance underwriters, which we estimate using sample data.
Recommended video:
04:48Population Standard Deviation Known
Sample Size
Sample size refers to the number of observations or data points collected from a population for analysis. In this case, the sample size is 32, which is crucial for calculating the confidence interval. A larger sample size generally leads to a more accurate estimate of the population mean and a narrower confidence interval.
Recommended video:
05:11Sampling Distribution of Sample Proportion
Related Practice
Textbook Question
Graphical Analysis In Exercises 9–12, use the values on the number line to find the sampling error.
77
views
Textbook Question
True or False? In Exercises 1 and 2, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
The point estimate for the population proportion of failures is 1-p^
88
views
Textbook Question
What happens to the shape of the chi-square distribution as the degrees of freedom increase?
222
views
Textbook Question
Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.
Social Security In a survey of 661 non-retired Americans, 218 said that they expect to rely on Social Security as major source of income when they retire. (Adapted from Gallup)
92
views
Textbook Question
For the same sample statistics, which level of confidence would produce the widest confidence interval? Explain your reasoning.
a. 90%
b. 95%
c. 98%
d. 99%
125
views