Question

In a random sample of 60 shoppers from a large suburban mall, 36 indicated that they had been to a movie in the past month. In an independent random sample of 50 shoppers chosen from the downtown shopping area, 31 indicated that they had been to a movie in the past month. What significance test should be used to determine whether these data provide sufficient evidence to reject the hypothesis that the proportion of suburban mall shoppers who had been to a movie in the same month is the same as the proportion downtown

Answer 1

Answer:

There is no enough statistical evidence to reject the hypothesis that the proportion of suburban mall shoppers who had been to a movie in the same month is the same as the proportion downtown.

Step-by-step explanation:

We have to perform an hypothesis test on the difference of proportions to see if there is enough evidence to reject the null hypothesis.

The null and alternative hypothesis are:

$H_0: \pi_1-\pi_2=0\\\\H_a: \pi_1-\pi_2\neq0$

The significance level is assumed to be α=0.05

The suburban sample has  a proportion of

$p_1=36/60=0.6$

The downtown sample has a proportion of

$p_2=31/50=0.62$

The average of proportion is:

$p=\frac{n_1p_1+n_2p_2}{n_1+n_2}=\frac{60*0.60+50*0.62}{60+50}=\frac{67}{110}=0.61$

The standard deviation of the difference of proportions is:

$s_{p1-p2}=\sqrt{\frac{p(1-p)}{n_1} +\frac{p(1-p)}{n_2} } =\sqrt{\frac{0.61*0.39}{60} +\frac{0.61*0.39}{50} }\\\\s_{p1-p2}=\sqrt{0.003965+0.004758}=0.0934$

Then, we can calculate the value of the statistic z

$z=\frac{p_1-p_2}{s}=\frac{0.60-0.62}{0.0934}=\frac{-0.02}{0.0934} = 0.214$

The P-value for this z is:

$P(|z|>0.214)=0.83054$

The P-value is bigger than the significance level, so there is no enough evidence to reject the null hypothesis.

Answer 2

Two proportion z test is used to compare two proportions. In the z test the null hypothesis is that the two proportions are same and the alternate hypothesis is that the proportions are not the same.For the given problem there is two proportion Z test test should be used to determine whether these data provide sufficient evidence to reject the hypothesis that the proportion of suburban mall shoppers who had been to a movie in the same month is the same as the proportion downtown

What is two proportion z test?

Two- proportion z test should be used to determine as the given data is sufficient evidence to reject the hypothesis that the proportion of shoppers at the suburban mall who had been to a movie in the same month is the same as the proportion downtown is equal to the proportion of shoppers in a large downtown shopping area who had been to a movie in the past month.

Two proportion z test-

• This test is used when the sample size is greater than the 30. In the given problem the sample size is greater than the 30.

• This test is performed when the data is normally distributed.

• This test is performed when the data is randomly selected.

All this condition is satisfied by the given problem.

Two proportion z test is used to compare two proportions. In the z test the null hypothesis is that the two proportions are same and the alternate hypothesis is that the proportions are not the same. The random sample of populations provide is same as the serve as two dimensions.

Hence, for the given problem there is two proportion Z test test should be used to determine whether these data provide sufficient evidence to reject the hypothesis that the proportion of suburban mall shoppers who had been to a movie in the same month is the same as the proportion downtown.

For more about the simple random sample test follow the link below-

https://brainly.com/question/2635297