MATH 140 -Problem Set 4A,4B,4C,4D,4E,4F,4G,4H

This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

Problems Section 4D

(#1-10) Use each of the following Goodness of Fit 𝜒𝜒2-test statistics and the corresponding critical values to fill out the table.

𝜒𝜒2-test stat

Sentence to explain 𝜒𝜒2-test statistic.

Critical Value

Does the 𝜒𝜒2-test statistic fall in a tail determined by the critical value?

(Yes or No)

Does sample data significantly

disagree with 𝐻𝐻0? 1. +28.573 +9.117 2. +1.226 +7.113 3. +2.137 +5.521 4. +14.415 +6.114 5. +3.718 +7.182 6. +0.891 +3.994 7. +51.652 +14.881 8. +1.185 +4.181 9. +2.442 +8.619

10. +14.133 +10.336 (#11-20) Use each of the following P-values and corresponding significance levels to fill out the table.

P-value Proportion

P- value

%

Sentence to explain the

P-value

Significance Level %

Significance level

Proportion

If 𝐻𝐻0 is true, could the sample data occur by random

chance or is it unlikely?

Reject 𝐻𝐻0 or Fail to reject 𝐻𝐻0?

11. 0.0006 10% 12. 0.042 1% 13. 9.16 × 10−7 5% 14. 0.739 1% 15. 0.0035 5% 16. 0 10% 17. 0.419 5% 18. 0.0274 10% 19. 3.77 × 10−5 1% 20. 0.067 5%

21. How is the degrees of freedom calculated in a Goodness of Fit test?

22. The 𝝌𝝌𝟐𝟐-test statistic compares the observed sample counts to the expected counts from 𝑯𝑯𝟎𝟎. Explain how the expected counts are calculated. 23. Explain how the 𝝌𝝌𝟐𝟐-test statistic is calculated from the observed and expected counts. 24. If the observed sample counts were significantly different from the expected counts, would the 𝜒𝜒2-test statistic be large or small? Explain why.

25. If the observed sample counts were close to the expected counts, would the 𝜒𝜒2-test statistic be large or small? Explain why.

This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

(#26-29) Directions: Use StatKey at www.lock5stat.com to simulate the following Chi-squared Goodness of Fit tests. Go to “more advanced randomization tests” at the bottom of the StatKey page. Click on the button that says “𝜒𝜒2 Goodness of Fit”. Under “Edit Data”, type in the given sample data. Create a randomized simulation of the null hypothesis to answer the following questions.

a) Write the null and alternative hypothesis. Include relationship implications.

b) What is the degrees of freedom?

c) What is the Chi-squared test statistic? Write a sentence to explain the test statistic.

d) Adjust the right tail of your simulation to reflect the significance level. Did the Chi-squared test statistic fall in the tail?

e) Does the sample data significantly disagrees with the null hypothesis? Explain your answer.

f) Are the observed counts in the sample data significantly different from the expected counts from the null hypothesis? Explain your answer.

g) Put the Chi-squared test statistic into the bottom box in the right tail of your simulation in order to calculate the P- value. What was the P-value? (Answers will vary.) Write a sentence to explain the P-value.

h) Use the P-value and significance level to determine if the sample data could have occurred by random chance (sampling variability) or is it unlikely to random chance? Explain your answer.

i) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer.

j) Write a conclusion for the hypothesis test. Explain your conclusion in plain language.

k) Is the population proportion related to the categorical variable or not? Explain your answer.

26. It is a big job to write and grade the AP-statistics exam for high school students each year. It is a difficult multiple-choice exam. All questions have five possible answers A-E. Use a 5% significance level and the following sample data to test the claim that percent of A answers is the same as the percent of B answers which is the same as C, D and E. This would indicate that the letter of the answer is not related to the percentage of times it happens. You can assume that the sample data meets the assumptions. Type the following sample data under the “Edit Data” menu of StatKey.

Choice, Count A, 85 B, 90 C, 79 D, 78 E, 68

This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

27. We collected data from all of the math 140 statistics students in the fall 2015 semester. A person that works at COC thinks that 80% of COC students drive alone, 10% carpool, 5% are dropped off by someone, 2% walk, 1% bike, and 2% use public transportation. Use a 5% significance level and the following sample data to test the claim that these percentages are wrong. You can assume that the data meets the assumptions for inference. Type in the proportions under “Null Hypothesis” in StatKey. Under “Edit Data” type in the following sample data from the fall 2015 survey data.

Choice, Count Bicycle, 1 Carpool, 30 Drive Alone, 267 Dropped Off, 18 Public Transportation, 6 Walk, 10

28. We collected data from all of the math 140 statistics students in the fall 2015 semester. Use a randomized simulation in StatKey, a 5% significance level, and the following sample data to test the claim that the population percentages for the different political parties are different. This would indicate that the political party is related to the population percentages. You can assume that the data meets the assumptions for inference. Under “Edit Data”, type in the following sample data from the fall 2015 survey data.

Choice, Count Democratic, 110 Republican, 63 Independent, 65 Other, 90

29. Juries are required to meet the racial demographic of the county they represent. Here is the racial demographic for Alameda county: 54% Caucasian, 18% African American, 12% Hispanic American, 15% Asian American, and 1% other. We are worried that the juries in Alameda County may not be representing these percentages. Use randomized simulation, a 1% significance level, and the following observed sample counts to test the claim that the juries do not represent the demographic of the county. Under the “Edit Data” menu in StatKey, type in the following sample counts.

Jury Sample Data Observed Counts

Race, Count Caucasian, 780 African American, 117 Hispanic American, 114 Asian American, 384 Other, 58

This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

Under the “Null Hypothesis” menu, type in the following.

(#30-32) Directions: Use the following Statcato printouts to answer the following questions.

a) Write the null and alternative hypothesis. Include relationship implications.

b) Check the assumptions for a Goodness of Fit test.

c) What is the Chi-squared test statistic? Write a sentence to explain the test statistic.

d) Did the Chi-squared test statistic fall in the tail determined by the critical value?

e) Does the sample data significantly disagrees with the null hypothesis? Explain your answer.

f) Are the observed counts in the sample data significantly different from the expected counts from the null hypothesis? Explain your answer.

g) What was the P-value? Write a sentence to explain the P-value.

h) Use the P-value and significance level to determine if the sample data could have occurred by random chance (sampling variability) or is it unlikely to random chance? Explain your answer.

i) Should we reject the null hypothesis or fail to reject the null hypothesis? Explain your answer.

j) Write a conclusion for the hypothesis test. Explain your conclusion in plain language.

k) Is the population proportion related to the categorical variable or not? Explain your answer.

This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

30. An online sports magazine wrote an article about the favorite sports in America. It said that 43% of Americans prefer Football, 23% of Americans prefer Baseball, 20% of Americans prefer Basketball, 8% of Americans prefer Hockey, and 6% of Americans prefer Soccer. When 130 randomly selected adults were asked their favorite sport, we found the following: 44 said Football, 26 said Baseball, 29 said Basketball, 13 said Hockey, and 18 said Soccer. Use a 5% significance level to test the claim that the proportions match the distribution claimed in the magazine article.

31. Thousands of people die from car accidents across the U.S. every year, but is the day of the week related to the probability of having a fatal car accident? To test this claim, use a 1% significance level and a Goodness of Fit test to determine if the probabilities of a fatal car accident are significantly different. The following random sample data summary gives the observed number of the number of deaths from car accidents in the U.S. for each day of a randomly selected week. The total number of deaths for the week was 805.

This chapter is from Introduction to Statistics for Community College Students, 1st Edition, by Matt Teachout, College of the Canyons, Santa Clarita, CA, USA, and is licensed

under a “CC-By” Creative Commons Attribution 4.0 International license – 10/1/18

32. The National Highway Traffic Safety Administration (NHTSA) publishes reports about motorcycle fatalities and helmet use. The following distribution shows the proportion of fatalities by location of injury for motorcycle accidents.

The random sample data below shows the distribution of 2068 randomly selected fatalities from riders that were not wearing a helmet. Use a 0.01 significance level to test the claim that the distribution for the sample does not match the proportions given by the NHTSA. Where is the largest discrepancy between the observed and expected value? What does this tell us about the importance of wearing helmets?

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