=====Mathematics 341 - Spring 2013====

====Problem Set # 5====

This set of problems is due at the beginning of class, Wednesday, 27 March.

Note: You do not have to show the R commands you use to solve the following problems. However, you do need to print out any graphical results, and label them appropriately. Moreover, for any hypothesis test, you must state the null hypothesis you are testing, define all parameters or random variables that you use, state the value of the test statistic, state the p-value for the test, and state what conclusion you may draw from the test. Report the expected frequencies for any goodness of fit tests.

Problems from Chapter 5: 5.23 (parts (a) and (e)), 5.27

And:

1. A study in 1984 examined the strength of Kevlar 49/epoxy, a material used in the space shuttle. The times to failure (in hours) of 101 strands at a stress level of 90% were found to be:

.01, .01, .02, .02, .02, .03, .03, .04, .05, .06, .07, .07, .08, .09, .09, .10, .10, .11, .11, .12, .13, .18, .19, .20, .23, .24, .24, .29, .34, .35, .36, .38, .40, .42, .43, .52, .54, .56, .60, .60, .63, .65, .67, .68, .72, .72, .72, .73, .79, .79, .80, .80, .83, .85, .90, .92, .95, .99, 1.00, 1.01, 1.02, 1.03, 1.05, 1.10, 1.10, 1.11, 1.15, 1.18, 1.20, 1.29, 1.31, 1.33, 1.34, 1.40, 1.43, 1.45, 1.50, 1.51, 1.52, 1.53, 1.54, 1.54, 1.55, 1.58, 1.60, 1.63, 1.64, 1.80, 1.80, 1.81, 2.02, 2.05, 2.14, 2.17, 2.33, 3.03, 3.03, 3.24, 4.20, 4.69, 7.89

Test how well an exponential distribution fits these data.

2. Yule (in 1900) presented the following table of data on the heights of 210 married couples:

^ ^  Tall wife  ^  Medium wife  ^  Short wife  ^
^Tall husband |  18|  28|  19|
^Medium husband |  20|  51|  28|
^Short husband  |  12|  25|  9|

Test the hypothesis that the heights of husbands and wives are independent.

3. A newspaper article in 1965 reported that a high school student observed 9207 heads and 8743 tails in 17,950 coin tosses.

(a) If $\pi$ is the probability of tossing a head, test the hypothesis that $H_0 : \pi = \frac{1}{2}$.

(b) It turned out that the student had tossed groups of five coins at a time, with the results as shown in the table below. Test the hypothesis that the observations are from a binomial distribution with probability of heads being $\pi = \frac{1}{2}$.

(%%c%%) Test the hypothesis that the observations are from a binomial distribution (that is, the coins each have the same probability of heads, but this probability is not necessarily $\frac{1}{2}$).

^  Number of Heads  ^  Frequency  ^
|  0  |  100|
|  1  |  524|
|  2  |  1080|
|  3  |  1126|
|  4  |  655|
|  5  |  105|