How influential can a movie be? A curious researcher asked a random group of peo

Question

How influential can a movie be? A curious researcher asked a random group of people aged 18 and over to rate the Bush administration on a scale from 1 to 10 (with 1 being the least supportive and 10 being most supportive). She then had them view the anti-Bush movie Fahrenheit 9/11 by Michael Moore after it had just been released and asked them the same question again. Given the results below, test the null hypothesis that there is no difference in the before and after support for the Bush administration. Before After 7 10 2 3 7 What is the observed t, df, and critical t? Do you accept or reject the null hypothesis? What do you conclude based on the results of the t test?

Explanation / Answer

The following table is obtained:

From the sample data, it is found that the corresponding sample means are:

X¯1=5.917

X¯2=3.75

Also, the provided sample standard deviations are:

s1=2.937

s2=2.598

and the sample size is n = 12. For the score differences we have

D¯=2.167

sD=1.467

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: D = 0

Ha: D 0

This corresponds to a two-tailed test, for which a t-test for two paired samples be used.

(2) Rejection Region

Based on the information provided, the significance level is =0.05, and the degrees of freedom are df=11.

Hence, it is found that the critical value for this two-tailed test is tc=2.201, for =0.05 and df=11.

The rejection region for this two-tailed test is R={t:|t|>2.201}.

(3) Test Statistics

The t-statistic is computed as shown in the following formula:

t=D¯/(sD/n) = 2.167/(1.467/12)=5.117

(4) Decision about the null hypothesis

Since it is observed that |t|=5.117>tc=2.201, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0003, and since p=0.0003<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean 1 is different than 2, at the 0.05 significance level.

Sample 1 Sample 2 Difference = Sample 1 – Sample 2 4 2 2 7 5 2 1 1 0 10 6 4 9 9 0 2 1 1 3 1 2 6 2 4 8 5 3 9 5 4 7 6 1 5 2 3 Average 5.917 3.75 2.167 St. Dev. 2.937 2.598 1.467 n 12 12 12

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