# 1.5: Chi-Square Practice

- Page ID
- 6136

## 11.1 Facts About the Chi-Square Distribution

1.

## 11.2 Test of a Single Variance

*Use the following information to answer the next three exercises:* An archer’s standard deviation for his hits is six (data is measured in distance from the center of the target). An observer claims the standard deviation is less.

6.

What type of test should be used?

State the null and alternative hypotheses.

**8**.

Is this a right-tailed, left-tailed, or two-tailed test?

Let \(\alpha = 0.05\)

Decision: ________________

Reason for the Decision: ________________

Conclusion (write out in complete sentences): ________________

**29**.

Does it appear that the pattern of AIDS cases in Santa Clara County corresponds to the distribution of ethnic groups in this county? Why or why not?

## 11.4 Test of Independence

*Determine the appropriate test to be used in the next three exercises.*

30.

A pharmaceutical company is interested in the relationship between age and presentation of symptoms for a common viral infection. A random sample is taken of 500 people with the infection across different age groups.

The owner of a baseball team is interested in the relationship between player salaries and team winning percentage. He takes a random sample of 100 players from different organizations.

**32**.

A marathon runner is interested in the relationship between the brand of shoes runners wear and their run times. She takes a random sample of 50 runners and records their run times as well as the brand of shoes they were wearing.

*Use the following information to answer the next seven exercises:* Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. Table \(\PageIndex{25}\) shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

Traveling distance | Third class | Second class | First class | Total |
---|---|---|---|---|

1–100 miles | 21 | 14 | 6 | 41 |

101–200 miles | 18 | 16 | 8 | 42 |

201–300 miles | 16 | 17 | 15 | 48 |

301–400 miles | 12 | 14 | 21 | 47 |

401–500 miles | 6 | 6 | 10 | 22 |

Total | 73 | 67 | 60 | 200 |

State the hypotheses.

\(H_0\): _______

\(H_a\): _______

**34**.

\(df\) = _______

How many passengers are expected to travel between 201 and 300 miles and purchase second-class tickets?

**36**.

How many passengers are expected to travel between 401 and 500 miles and purchase first-class tickets?

**37**.

What is the test statistic?

**38**.

What can you conclude at the 5% level of significance?

*Use the following information to answer the next eight exercises:* An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.

**39**.

Complete the table.

Smoking level per day | African American | Native Hawaiian | Latino | Japanese Americans | White | Totals |
---|---|---|---|---|---|---|

1-10 | ||||||

11-20 | ||||||

21-30 | ||||||

31+ | ||||||

Totals |

State the hypotheses.

\(H_0\): _______

\(H_a\): _______

**41**.

Enter expected values in Table \(\PageIndex{26}\). Round to two decimal places.

Calculate the following values:

\(df\) = _______

**43**.

\(\chi^2\) test statistic = ______

**44**.

Is this a right-tailed, left-tailed, or two-tailed test? Explain why.

**45**.

Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the confidence level.

State the decision and conclusion (in a complete sentence) for the following preconceived levels of *\alpha*.

**46**.

\(\alpha = 0.05\)

- Decision: ___________________
- Reason for the decision: ___________________
- Conclusion (write out in a complete sentence): ___________________

**47**.

\(\alpha = 0.01\)

## 11.5 Test for Homogeneity

**48**.

A math teacher wants to see if two of her classes have the same distribution of test scores. What test should she use?

What are the null and alternative hypotheses for Table \(\PageIndex{27}\).

20–30 | 30–40 | 40–50 | 50–60 | |
---|---|---|---|---|

Private practice | 16 | 40 | 38 | 6 |

Hospital | 8 | 44 | 59 | 39 |

State the null and alternative hypotheses.

**54**.

\(df\) = _______

**55**.

What is the test statistic?

What can you conclude at the 5% significance level?

## 11.6 Comparison of the Chi-Square Tests

**57**.

Which test do you use to decide whether an observed distribution is the same as an expected distribution?

What is the null hypothesis for the type of test from **Exercise \(\PageIndex{57}\)**?

**59**.

Which test would you use to decide whether two factors have a relationship?

Which test would you use to decide if two populations have the same distribution?

**61**.

How are tests of independence similar to tests for homogeneity?

**62**.

How are tests of independence different from tests for homogeneity?