9.1 (a–b)

9.3 The conditional distribution for whether the wallet contained money or did not is more informative for comparing how many were returned or not based on the wallet’s contents.

9.7 They are shown in the following table.

9.9

1. df=9, from table, 0.10<P-value<0.15; using software, P-value=0.1413.

2. df=4, from table, 0.005<P-value<0.01; from software, P-value=0.0091.

• (c–d) df=2, from table, 0.025<P-value<0.05; from software, P-value=0.0450.

9.11

1. Explanatory 1: 34.3% Yes and 65.7% No. Explanatory 2: 45.7% Yes and 54.3% No.

2. Explanatory variable value 1 had proportionately fewer “yes” responses.

9.13

1. pi=proportion of “yes” responses in group i. H0: p1=p2, Ha: p1≠p2; p^1=0.343; p^2=0.457,

2. z=−2.39, P-value<0.0168.

3. The P-values agree.

4. z2=(-2.39)2=5.714.

9.15 This is due to rounding error.

9.17 California: 0.5820, Hawaii: 0.0000, Indiana: 0.0196, Nevada: 0.0660, Ohio: 0.2264; 0.0369+0.5820+0.0000+0.0196+0.0660+0.2264=0.931.

9.1

1. Age is explanatory; Rejection is the response variable.

2. Joint distribution

   Rejected	Age
   	Under 20	20 to 25	25 to 30	30 to 35	35 to 40	Over 40
   Yes	0.0002	0.0019	0.0033	0.0053	0.0086	0.0114
   No	0.1761	0.2333	0.1663	0.1316	0.1423	0.1196

3. Marginal distribution of Age

   Age
   Under 20	20 to 25	25 to 30	30 to 35	35 to 40	Over 40
   0.1763	0.2352	0.1696	0.1369	0.1509	0.1310

   Marginal distribution of Rejected

   Rejected
   Yes	0.0308
   No	0.9692

4. Conditional distribution based on Age

   	Age
   Rejected	Under 20	20 to 25	25 to 30	30 to 35	35 to 40	Over 40
   Yes	0.0012	0.0082	0.0196	0.0389	0.0572	0.0868
   No	0.9988	0.9918	0.9804	0.9611	0.9428	0.9132

   Conditional distribution based on Rejected

   	Age
   Rejected	Under 20	20 to 25	25 to 30	30 to 35	35 to 40	Over 40
   Yes	0.0066	0.0628	0.1082	0.1731	0.2803	0.3690
   No	0.1817	0.2407	0.1716	0.1658	0.1468	0.1234

9.3 Answers will vary.

9.5 Table of expected counts

9.7 X2=9194.7, df=5, P-value<0.0001. There is a relationship between age and whether someone was rejected from the Spanish American War.

9.9 X2=88.1376 approximately equal to z2=(−9.39)2=88.1721.

9.11 More and Boys: 37.23%; More and Girls: 34.13%. Never and Boys: 5.39%; Never and Girls: 7.12%. Once and Boys: 6.36%; Once and Girls: 9.77%. 48.98% are boys, 51.02% are girls. 71.36% have witnessed it more than once, 16.12% have witnessed it once, and 12.51% have never witnessed it. For boys: 76.01% have witnessed it more than once, 12.98% have witnessed it once, and 11.01% have never witnessed it. For girls: 66.90% have witnessed it more than once, 19.14% have witnessed it once, and 13.96% have never witnessed it. For those witnessing sexual harassment more than once: 52.17% are boys, 47.83% are girls. For those witnessing sexual harassment once: 39.43% are boys, 60.57% are girls. For those never witnessing sexual harassment: 43.09% are boys, 56.91% are girls. The distribution of times by sex are probably the most informative, showing that boys tend to say they have witnessed sexual harassment slightly more than girls, with boys having 10% higher in the More category than the girls.

9.13 X2=20.822, df=2, P-value<0.0001.

9.15

1. The null hypothesis is that the coin is fair. The alternative is that the coin is biased.

2. The expected counts would be 5000 heads and 5000 tails.

9.17 X2=1.7956. df=1, 0.15<P-value<0.20. Fail to reject the null. There is no evidence that the coin is biased.

9.19 Answers will vary.

9.21 X2=15.28, df=2, P-value<0.0005. Fail to reject the null hypothesis. These observed counts do not suggest that they follow the Poisson distribution.

9.23

1. H0: p1=p2, Ha: p1≠p2; p^1=0.8892, p^2=0.3120, p^=0.5200, z=17.557, P-value≈0.

2. H0: There is no association between being harassed online and in person, Ha: There is a relationship; X2=308.23, df=1, P-value≈0.

3. 17.5572=308.25, which agrees with X2.

4. Perhaps one girl wouldn’t answer these questions.

9.25

1. The solution to Exercise 9.23 used “harassed online” as the explanatory variable.

2. Changing to use “harassed in person” for the two-proportions z test gives p^1=0.6161, p^2=0.0832, p^=0.3603. We again compute z=17.557, P-value≈0. No changes will occur in the chi-square test.

3. The test statistic will be the same regardless of which is viewed as explanatory.

9.27 Expected counts are 83.3 for each outcome.

9.29 H0: the DFW rate has not changed; X2=308.3, df=2, P<0.0001.

9.31

1. For example, among those students in trades, 320.28 enrolled right after high school, and 621.72 enrolled later.

2. For example, 39.4% of these students enrolled right after high school. Health is the most popular field, with 38%.

3. X2=276.1, df=5, P-value<0.0001.

9.33 X2=852.4330, df=1, P-value<0.0001, z2=(−29.2)2=852.64=X2 with rounding error.

9.35 Answers will vary based on the data set generated. Most results will give a fairly decent randomization and should fail to reject the null hypothesis. Changing the interval likely will not change the result and should still fail to reject the null hypothesis.

9.37 The expected counts are both 120. X2=106.67, df=1, P-value<0.0001. We conclude that the proportion who were harassed online and the proportion who were harassed in person are different.

9.39

1. 	Stratum
   Claim	Small	Medium	Large	Total
   Allowed	52	13	2	67
   Not	7	6	2	15
   Total	59	19	4

2. 11.86% of Small, 31.58% of Medium, and 50% of Large were not allowed.

3. H0: There is no association between the size of the claim and whether it is allowed, Ha: There is a relationship.

4. X2=6.566, df=2, 0.01<P-value<0.05. There is evidence that there is an association between the size of claim and whether it is allowed.

5. The expected counts for the large companies are small so the chi-square statistic may not be reliable.