Department of Statistics, Virginia Tech

This homework is due on **Thursday, October 26th at 2pm** (the start of class). Please turn in all your work. This homework primarily covers \(r \times c\) contingency tables, median test, goodness-of-fit test, contingency coefficients and Cochran’s test.

**Calculations by hand**: Throughout this homework, and beyond, “by hand” means either (1) you utilize quantile/distribution tables, and/or Gaussian approximations, as appropriate, and otherwise do all of your calculations with pen and paper (and a calculator); or (2) you write code, say in R, building up all of the steps yourself, i.e., not using a library function that automates the entire procedure (see next bullet).**Using a software library**: Through this the homework, and beyond, “using a software library” means you can feed your data into a built-in function, like`t.test`

and`binom.test`

in R, and interpret the output as appropriate. Be sure to provide details on the library you used, how you used it, what the output was, and what it means.

Three professors are teaching large classes in introductory statistics. At the end of the semester, they compare grades to see if there are significant differences in their grading policies. Are these differences significant?

Prof | A | B | C | D | F | WP | WF |
---|---|---|---|---|---|---|---|

Smith |
12 | 45 | 49 | 6 | 13 | 18 | 2 |

Jones |
10 | 32 | 43 | 18 | 4 | 12 | 6 |

White |
15 | 19 | 32 | 20 | 6 | 9 | 7 |

- (3 pts) Which test are you using?
- (7 pts) Perform the appropriate test “by hand”.
- (5 pts) Perform the test in (ii) “using a software library”.

A random sample of 30 graduating university seniors was categorized by college and religious preference as follows. Is there a relationship between college and religious preference?

Arts & Sciences | Business | Engineering | Other |
---|---|---|---|

Catholic | Protestant | Catholic | Protestant |

Catholic | Catholic | Protestant | Catholic |

Jewish | Jewish | Protestant | Catholic |

Catholic | Jewish | Protestant | Other |

Protestant | Protestant | Other | Catholic |

Protestant | Protestant | Other | |

Other | Other | Catholic | |

Protestant | Jewish | ||

Other | Other |

- (3 pts) Which test are you using?
- (7 pts) Perform the appropriate test “by hand”.
- (5 pts) Perform the test in (ii) “using a software library”.

An experiment was conducted to determine the compression strength (lb) for different types of boxes. We are interested in determining if the median is the same for all types of boxes.

Box 1 |
655.5 | 788.3 | 734.3 | 721.4 |

Box 2 |
789.2 | 772.5 | 786.9 | 686.1 |

Box 3 |
737.1 | 539.0 | 696.3 | 671.7 |

Box 4 |
535.1 | 628.7 | 542.4 | 559.0 |

- (10 pts) Perform the test “by hand” including pairwise comparisons if appropriate.
- (5 pts) Perform the test “using a software library”.

A random sample of 30 stocks was selected from each of the three major U.S exchanges, and their performance over the previous year was monitored. The median performance for all 90 stocks was noted and the following table constructed:

Exchange | Above Median |
---|---|

New York | 18 |

American | 17 |

NASDAQ | 10 |

Was there a significant difference in the performance of stocks on the three exchanges during the previous year?

- (8 pts) Perform the test “by hand”.
- (7 pts) Perform the test “using a software library”.

A certain type of insect that is found in lakes in the southwestern United States is studied to see if the chromosomal structure is significantly different among states. The number of insects of various chromosomal types is recorded as follows:

Type | Texas | New Mexico | Arizona | California |
---|---|---|---|---|

A |
54 | 72 | 83 | 96 |

B |
20 | 6 | 18 | 6 |

C |
17 | 3 | 12 | 0 |

D |
0 | 12 | 14 | 1 |

E |
0 | 10 | 0 | 0 |

- (10 pts) Compute the following “by hand”:
- (3 pts) \(T\).
- (2 pts) \(R_1\).
- (1 pts) Cramer’s contingency coefficient.
- (2 pts) Pearson’s contingency coefficient.
- (2 pts) Pearson’s Mean-Square contingency coefficient.

- (5 pts) Find the available coefficients “using a software library”.

A die was cast 600 times with the following results:

Occurrence |
1 | 2 | 3 | 4 | 5 | 6 |

Frequency |
87 | 96 | 108 | 89 | 122 | 98 |

Is the die balanced?

- (5 pts) Perform the test by hand.
- (5 pts) Perform the test using software.

In an attempt to compare the relative power of three statistical tests, 100 sets of artificial data were generated using a computer. On each set of data the three statistical tests were used with \(\alpha=0.05\), and the decision to accept or reject the null hypothesis was recored. The results are as follows:

Test 1 | Test 2 | Test 3 | Numbers of Sets of Data |
---|---|---|---|

Accept | Accept | Accept | 26 |

Accept | Accept | Reject | 6 |

Accept | Reject | Accept | 12 |

Reject | Accept | Accept | 4 |

Reject | Reject | Accept | 18 |

Reject | Accept | Reject | 5 |

Accept | Reject | Reject | 7 |

Reject | Reject | Reject | 22 |

Is there a difference in the power of the three tests when applied to populations from which the simulated data were obtained?

- (3 pts) Which test are you using?
- (7 pts) Perform the appropriate test “by hand”.
- (5 pts) Perform the test in (ii) “using a software library”.