### Who knows about the t test for a single sample

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1. Two boats, the Prada (Italy) and the Oracle (USA), are competing for a spot in the upcoming America's Cup race. They race over a part of the course several times. The sample times in minutes for the Prada were: 12.9, 12.5, 11.0, 13.3, 11.2, 11.4, 11.6, 12.3, 14.2, and 11.3. The sample times in minutes for the Oracle were: 14.1, 14.1, 14.2, 17.4, 15.8, 16.7, 16.1, 13.3, 13.4, 13.6, 10.8, and 19.0. For data analysis, the appropriate test is the t-Test: Two-Sample Assuming Unequal Variances.

The next table shows the results of this independent t-test. At the .05 significance level, can we conclude that there is a difference in their mean times? Explain these results to a person who knows about the t test for a single sample but is unfamiliar with the t test for independent means.

 Hypothesis Test: Independent Groups (t-test, unequal variance) Prada Oracle 12.170 14.875 mean 1.056 2.208 std. dev. 10 12 n 16 df -2.7050 difference (Prada - Oracle) 0.7196 standard error of difference 0 hypothesized difference -3.76 t .0017 p-value (two-tailed) -4.2304 confidence interval 95.% lower -1.1796 confidence interval 95.% upper 1.5254 margin of error

2. The Willow Run Outlet Mall has two Haggar Outlet Stores, one located on Peach Street and the other on Plum Street. The two stores are laid out differently, but both store managers claim their layout maximizes the amounts customers will purchase on impulse. A sample of ten customers at the Peach Street store revealed they spent the following amounts more than planned: \$17.58, \$19.73, \$12.61, \$17.79, \$16.22, \$15.82, \$15.40, \$15.86, \$11.82, \$15.85. A sample of fourteen customers at the Plum Street store revealed they spent the following amounts more than they planned when they entered the store: \$18.19, \$20.22, \$17.38, \$17.96, \$23.92, \$15.87, \$16.47, \$15.96, \$16.79, \$16.74, \$21.40, \$20.57, \$19.79, \$14.83. For Data Analysis, a t-Test: Two-Sample Assuming Unequal Variances was used.

At the .01 significance level is there a difference in the mean amount purchased on an impulse at the two stores? Explain these results to a person who knows about the t test for a single sample but is unfamiliar with the t test for independent means.

 Hypothesis Test: Independent Groups (t-test, unequal variance) Peach Street Plum Street 15.8680 18.2921 mean 2.3306 2.5527 std. dev. 10 14 n 20 df -2.42414 difference (Peach Street - Plum Street) 1.00431 standard error of difference 0 hypothesized difference -2.41 t .0255 p-value (two-tailed) -5.28173 confidence interval 99.% lower 0.43345 confidence interval 99.% upper 2.85759 margin of error

3. Fry Brothers heating and Air Conditioning, Inc. employs Larry Clark and George Murnen to make service calls to repair furnaces and air conditioning units in homes. Tom Fry, the owner, would like to know whether there is a difference in the mean number of service calls they make per day. Assume the population standard deviation for Larry Clark is 1.05 calls per day and 1.23 calls per day for George Murnen. A random sample of 40 days last year showed that Larry Clark made an average of 4.77 calls per day. For a sample of 50 days George Murnen made an average of 5.02 calls per day. At the .05 significance level, is there a difference in the mean number of calls per day between the two employees? What is the p-value?

 Hypothesis Test: Independent Groups (t-test, pooled variance) Larry George 4.77 5.02 mean 1.05 1.23 std. dev. 40 50 n 88 df -0.25000 difference (Larry - George) 1.33102 pooled variance 1.15370 pooled std. dev. 0.24474 standard error of difference 0 hypothesized difference -1.02 t .3098 p-value (two-tailed) -0.73636 confidence interval 95.% lower 0.23636 confidence interval 95.% upper 0.48636 margin of error

4. A consumer organization wants to know if there is a difference in the price of a particular toy at three different types of stores. The price of the toy was checked in a sample of five discount toy stores, five variety stores, and five department stores. The results are shown below.

 Discount toy Variety Department \$12 15 19 13 17 17 14 14 16 12 18 20 15 17 19

An ANOVA was run and the results are shown below. At the .05 significance level, is there a difference in the mean prices between the three stores? What is the p-value? Explain why an ANOVA was used to analyze this problem.

 One factor ANOVA Mean n Std. Dev 13.2 5 1.30 Discount Toys 16.2 5 1.64 Variety 18.2 5 1.64 Department 15.9 15 2.56 Total ANOVA table Source SS df MS F p-value Treatment 63.33 2 31.667 13.38 .0009 Error 28.40 12 2.367 Total 91.73 14

5. A physician who specializes in weight control has three different diets she recommends. As an experiment, she randomly selected 15 patients and then assigned 5 to each diet. After three weeks the following weight losses, in pounds, were noted. At the .05 significance level, can she conclude that there is a difference in the mean amount of weight loss among the three diets?

 Plan A Plan B Plan C 5 6 7 7 7 8 4 7 9 5 5 8 4 6 9

An ANOVA was run and the results are shown below. At the .01 significance level, is there a difference in the weight loss between the three plans? What is the p-value? What can you do to determine exactly where the difference is?

 One factor ANOVA Mean n Std. Dev 5.0 5 1.22 Plan A 6.2 5 0.84 Plan B 8.2 5 0.84 Plan C 6.5 15 1.64 Total ANOVA table Source SS df MS F p-value Treatment 26.13 2 13.067 13.52 .0008 Error 11.60 12 0.967 Total 37.73 14

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