Question

A filmmaker is considering two possible endings to her film. She shows both versions of her film to 50 young adults (aged 18-30) and 60 older adults (aged 30-50). Of the young adults, 39 preferred the first ending. Of the older adults, 43 preferred the first ending. She is interested in whether the proportions of individuals who prefer the first ending differ between the two age groups.

a. Which assumption is required for the filmmaker to be able to address her question of interest?

b. What are the null and alternative hypotheses?

c. What is the value of the test statistic?

d. What is the p-value? Give an expression involving a probability, not just a final answer.

e. State your conclusions in the language of the problem. Use a significance level of 5%.

f. Give a 95% confidence interval for the difference between the proportion of younger adults who prefer the first ending and the proportion of older adults who prefer the first ending.

Answer 1

a) Standard deviation of the two sample proportion is same

b)  the null hypothesis ( H₀ ) and alternative hypotheses (H_a ) are as follows :

Let's used minitab:

Step 1: Click on Stat >>> Basic Statistics >>>2 Proportions...

Step 2: Select Summarized data

Fill the given information

Look the following picture ...

Then click on Option:

Look the following image:

Then click on OK again click on Ok

So we get the following output

From the above output

c ) the z test statistics = 0.76 ,

d p-value = 0.448

We find z test statistic as 0.76

p-value for two tailed test is 2*P( Z < -0.76) this is the expression of p-value.

e. Determine the rejection rule:

Decision rule: 1) If p-value < level of significance (alpha) then we reject null hypothesis

2) If p-value > level of significance (alpha) then we fail to reject null hypothesis.

Here p value = 0.448 > 0.05 so we used 2nd rule.

That is we fail to reject null hypothesis

e ) Conclusion: At 5% level of significance there are not sufficient evidence to say that the proportions of individuals who prefer the first ending differ between the two age groups.

f )  The 95% confidence interval for the difference between the proportion of younger adults who prefer the first ending and the proportion of older adults who prefer the first ending is (-0.098, 0.225)