Question

A school psychologist wishes to determine whether a new anti-smoking film actually reduces the daily consumption of cigarettes by teenage smokers. The mean daily cigarette consumption is calculated for each of eight teenage smokers during the month before and the month after the film presentation, with the following results: MEAN DAILY CIGARETTE CONSUMPTION

SMOKER NUMBER BEFORE FILM (X1) AFTER FILM (X2)

1 28 26

2 29 27

3 31 32

4 44    44

5 35 35

6 20 16

7 50 47

8 25 23

A) Is there a significant difference in the number of cigarettes smoked before the film as compared to the number of cigarettes smoked after the film?

B) What does this NOT necessarily mean?

C) What might be done to improve the design of this experiment?

Answer 1

sample 1 is before X1 and sample 2 is after X2

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

Xˉ1 =32.75 and Xˉ2 =31.25
Also, the provided sample standard deviations are:

s1=9.939 and s2 =10.498
and the sample size is n = 8. For the score differences we have

Dˉ=1.5 and sD =1.69
(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:μD= 0

Ha:μD ≠ 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 α=0.05, and the degrees of freedom are df=7.

Hence, it is found that the critical value for this two-tailed test is tc =2.365, for α=0.05 and df=7.

The rejection region for this two-tailed test is R={t:∣t∣>2.365}.

(3) Test Statistics

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

t= Dbar/sd/sqrt(n)=2.51
(4) Decision about the null hypothesis

Since it is observed that ∣t∣=2.51>t c =2.365, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0404, and since p = 0.0404 & p=0.0404<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 μ1 is different than μ 2, at the 0.05 significance level.

c) Design can be more randomized and more data can be add for more precision.