Textbook Question
In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.
c = 0.95, σ = 2.5, E = 1.
100
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 6.4.8
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 = 0.80, n = 51
Verified step by step guidance1
Determine the degrees of freedom (df) using the formula: df = n - 1, where n is the sample size. In this case, df = 51 - 1.
Identify the level of confidence (c) and calculate the significance level (α) using the formula: α = 1 - c. For c = 0.80, α = 1 - 0.80.
Divide the significance level (α) into two tails for a two-tailed test. The left tail will have α/2, and the right tail will also have α/2.
Use a chi-square distribution table or statistical software to find the critical values χL² and χR². For χL², find the value corresponding to the cumulative probability of α/2 with df degrees of freedom. For χR², find the value corresponding to the cumulative probability of 1 - α/2 with df degrees of freedom.
Verify the critical values by ensuring they correspond to the correct cumulative probabilities and degrees of freedom in the chi-square distribution table or software.
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.
Chi-Square Distribution
The Chi-Square distribution is a statistical distribution that is used primarily in hypothesis testing and confidence interval estimation for categorical data. It is defined by its degrees of freedom, which are determined by the sample size and the number of parameters being estimated. The distribution is positively skewed, and as the degrees of freedom increase, it approaches a normal distribution.
Recommended video:
Guided course
07:01
Intro to Least Squares Regression
Critical Values
Critical values are the threshold points that define the boundaries of the acceptance region in hypothesis testing. They are determined based on the desired level of confidence (c) and the specific distribution being used, such as the Chi-Square distribution. For a given confidence level, critical values help in deciding whether to reject or fail to reject the null hypothesis.
Recommended video:
05:50Critical Values: t-Distribution
Degrees of Freedom
Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation. In the context of the Chi-Square distribution, degrees of freedom are typically calculated as the sample size minus one (n - 1) for a single sample. They play a crucial role in determining the shape of the Chi-Square distribution and the corresponding critical values.
Recommended video:
05:50Critical Values: t-Distribution
Related Practice
Textbook Question
In Exercises 9–12, construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed.
c = 0.98, xbar = 4.3, s = 0.34, n = 14
107
views
Textbook Question
Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.
c = 0.88
101
views
Textbook Question
In Exercises 13 and 14, use the confidence interval to find the margin of error and the sample mean.
(14.7, 22.1)
95
views
Textbook Question
In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.
c = 0.90, σ = 6.8, E = 1.
148
views
Textbook Question
Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.
In a survey of 220 U.S. adults ages 18–29, 65% said that they use Snapchat. The survey’s margin of error is ±7.9%. (Source: Pew Research Center)
73
views