## Instructions

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.

### Problem 1: Grading policies (15 pts)

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
1. (3 pts) Which test are you using?
2. (7 pts) Perform the appropriate test “by hand”.
3. (5 pts) Perform the test in (ii) “using a software library”.

### Problem 2: Religious preferences (15 pts)

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
1. (3 pts) Which test are you using?
2. (7 pts) Perform the appropriate test “by hand”.
3. (5 pts) Perform the test in (ii) “using a software library”.

### Problem 3: Compression strength (15 pts)

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
1. (10 pts) Perform the test “by hand” including pairwise comparisons if appropriate.
2. (5 pts) Perform the test “using a software library”.

### Problem 4: Stock performance (15 pts)

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?

1. (8 pts) Perform the test “by hand”.
2. (7 pts) Perform the test “using a software library”.

### Problem 5: Chromosomal structure (15 pts)

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
1. (10 pts) Compute the following “by hand”:
1. (3 pts) $$T$$.
2. (2 pts) $$R_1$$.
3. (1 pts) Cramer’s contingency coefficient.
4. (2 pts) Pearson’s contingency coefficient.
5. (2 pts) Pearson’s Mean-Square contingency coefficient.
2. (5 pts) Find the available coefficients “using a software library”.

### Problem 6: Rolling dice (10 pts)

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?

1. (5 pts) Perform the test by hand.
2. (5 pts) Perform the test using software.

### Problem 7: Power of tests (15 pts)

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?

1. (3 pts) Which test are you using?
2. (7 pts) Perform the appropriate test “by hand”.
3. (5 pts) Perform the test in (ii) “using a software library”.