A study of tipping behaviors examined the relationship between the color of the shirt worm by the server and whether or not the customer left a tip. There were 418 male customers in the study; 40 of the 69 who were served by a server wearing a red shirt left a tip. Of the 349 who were served by a server wearing a different colored shirt, 130 left a tip.

a. find the large-sample 95% confidence interval for the difference in proportions and use the scenario to explain the meaning of the confidence interval.

b. perform the large-sample significance test and use the scenario to explain the meaning of the significance test.

Step-by-step explanation:

Here, 40 of the 69 who were served by a server wearing a red shirt left a tip. Of the 349 who were served by a server wearing a different colored shirt, 130 left a tip.

Therefore:

2.1 40 72 130

The sample sizes are:

721 69 722 349 2

Two proportions and their difference are:

400.580 0.580 p = =_= 69 130 1300.372 D =-= n2 349 p1-P2 = 0.580-0.372 = 0.208

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c) For 95% confidence level, critical value of z is 1.96.

The large-sample 95% confidence interval for the difference in proportions is:

n1 7l n2 0.580(1 - 0.580) 0.372(1- 0.372) 0.208 ± 1.96 0.208 t 0.127 or (0.081,0.335)

We are 95% confident that the difference in proportion of male customers left a tip who were served by a server wearing a red shirt and who were served by a server wearing a different colored shirt lies between 0.081 and 0.335. Since 0 does not lie in the confidence interval, we can conclude that higher proportion of male customers left a tip who were served by a server wearing a red shirt than those who were served by a server wearing a different colored shirt.

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d) The hypotheses are:

\\H_0:p_1=p_2 \\H_a:p_1\ne p_2

The pooled proportion is:

1240 +130 +n2 69+34 0.4067

The test statistic is:

pi-2 0.580 0.372 V mi-P)(4+4) ν/0 4067(1-04067) (かー = 3.20 ) 0.4067(1-0.4067) (69 т 349

The p-value is:

p-value = P(z <-3.20) + P(z > 3.20) = 0.0007 0.0007 = 0.00 1 4

Since p-value is less than 0.05, reject the null hypothesis. We can conclude that there is significant difference in proportion of male customers left a tip who were served by a server wearing a red shirt and who were served by a server wearing a different colored shirt.

Please see attachment for better indentation and formula input in the solution.