Question

In a random sample of 30 people who rode a roller coaster one day, the mean wait time is 46.7 minutes with a standard deviation of 9.2 minutes. In a random sample of 50 people who rode a Ferris wheel the same day, the mean wait time is 13.3 minutes with a standard deviation of 1.9 minutes. Construct a 99% confidence interval for the difference between the mean wait times of everyone who rode both rides.
A. (31.7, 35.1)
B. (30.5, 36.3)
C. (29, 37.8)
D. (28.7, 38.1)

Answer 1

Answer: C. $(29,\ 37.8)$

Step-by-step explanation:

The confidence interval for difference of two population mean is given by :-

$\overline{x}_1-\overline{x}_2\pm z_{\alpha/2}\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}$

Given : Level of significance : $1-\alpha:0.99$

Then , significance level : $\alpha: 1-0.99=0.01$

Critical value : $z_{\alpha/2}=z_{0.005}=2.576$

$n_1=30\ ;\ n_2=50\\\\\overline{x}_1=46.7\ ;\ \overline{x}_2=13.3\\\\s_1=9.2\ ;\ s_2=1.9$

$46.7-13.3\pm(2.576)\sqrt{\dfrac{9.2^2}{30}+\dfrac{1.9^2}{50}}\approx33.4\pm4.38=(29.02\ ,37.78)\approx(29,\ 37.8)$

Hence,  a 99% confidence interval for the difference between the mean wait times of everyone who rode both rides $(29,\ 37.8)$