Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

Statistics

9. Hypothesis Testing for One Sample

Steps in Hypothesis Testing

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1:01 minutes

Problem 9.4.1d

Textbook Question

F Test Statistic

d. Is the F distribution symmetric, skewed left, or skewed right?

Verified step by step guidance

Understand the F distribution: The F distribution is used in statistics to compare variances between two groups or to test hypotheses in analysis of variance (ANOVA). It is defined by two degrees of freedom parameters, one for the numerator and one for the denominator.

Recognize the shape of the F distribution: The F distribution is not symmetric. It is skewed to the right, meaning it has a longer tail on the right side of the distribution.

Consider why the F distribution is skewed: The skewness arises because the F statistic is a ratio of two independent chi-squared distributions divided by their respective degrees of freedom. Ratios of squared values tend to produce a distribution with a right skew.

Note the range of the F distribution: The F statistic can only take positive values (greater than or equal to 0), as variances and squared values cannot be negative. This contributes to the right skew of the distribution.

Apply this knowledge in practice: When interpreting results involving the F distribution, remember its skewed nature and the fact that larger F values indicate greater evidence against the null hypothesis in hypothesis testing scenarios.

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Key Concepts

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

F Distribution

The F distribution is a probability distribution that arises frequently in statistics, particularly in the context of variance analysis. It is defined by two parameters, the degrees of freedom for the numerator and the denominator, which correspond to the variances being compared. The F distribution is used to determine if there are significant differences between the variances of two or more groups.

Skewness

Skewness refers to the asymmetry of a probability distribution. A distribution is said to be skewed left (negatively skewed) if it has a longer tail on the left side, while it is skewed right (positively skewed) if the tail is longer on the right side. Understanding skewness is crucial for interpreting the shape of the distribution and its implications for statistical analysis.

Properties of the F Distribution

The F distribution is inherently right-skewed, meaning it has a longer tail on the right side. This characteristic is important when conducting hypothesis tests, as it affects the critical values used to determine statistical significance. The right skewness indicates that most of the values are concentrated on the left, with fewer extreme values on the right, which is typical in variance comparisons.

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Textbook Question

Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

b. Use the P-value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

Textbook Question

F Test Statistic

a. If s2,1 represents the larger of two sample variances, can the F test statistic ever be less than 1?

Textbook Question

F Test Statistic

b. Can the F test statistic ever be a negative number?

Textbook Question

F Test Statistic

c. If testing the claim that sigma2,1 is not equals to sigma2,2 what do we know about the two samples if the test statistic F is very close to 1?

Textbook Question

Bootstrapping and Randomization When resampling data from two independent samples, what is the fundamental difference between bootstrapping and randomization?

Textbook Question

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

a. What does the symbol y^ represent?

Textbook Question

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

c. What is the predictor variable?

Textbook Question

Notation What is the difference between the regression equation y^ = b0 + b1x and the regression equation y = β0 + β1x.

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