Question

The Mungo Opinion Research Organization assigned a pollster to collect data from 30 randomly selected adults. As the data were   submitted to the company, the gender of each interviewed participant was noted. The sequence obtained is shown in the accompanying list.

M     M     F     M     M     F     M     M     F     M     F     M     M     F     M

F      M      F     M     M     F     M     M     F     M     M     F     M     M     M                              

1. At the 0.05 significance level, test the claim that the sequence is random.

2. At the 0.05 significance level, test the claim that the proportion of women is different from 0.5

3. Use the sample to construct a 95% confidence interval for the proportion of women.

4. What do the preceding results suggest? Is the sample biased against women? Was the sample obtained in a random sequence? If you are the manager, will you have any problems with these results?

Answer 1

1.

We will use run test for Detecting Non-randomness.

H0:   the sequence of men and women was produced in a random manner
Ha:   the sequence of men and women was not produced in a random manner

The test statistic is
Z = (R - R1) / Sr

where R is the observed number of runs, R1, is the expected number of runs, and Sr is the standard deviation of the number of runs. The values of R1 and Sr are computed as follows:

with n1 and n2 denoting the number of women and men values in the series.

Here n1 = 10, n2 = 20

Sr = 2.38

From the sequence, the observed number of runs is 21.

Z = (R - R1) / Sr = (21 - 14.33) / 2.38 = 2.80

For 0.05 significance level, the critical value of Z is 1.96. As the test statistic Z (2.80) is greater than the critical value, we reject the null hypothesis and conclude that there is significant evidence that the sequence of men and women was not produced in a random manner.

2.

H0:   the proportion of women is 0.5. That is p = 0.5
Ha:   the proportion of women is not 0.5. That is p 0.5

Sample proportion = 10 / 30 = 0.3

Standard error of proportion =

Test statistic, Z = (sample proportion - hypothesized proportion) / SE

= (0.3 - 0.5) / 0.0913

= -2.19

For 0.05 significance level, the critical value of Z is 1.96. As the absolute value of test statistic Z (2.19) is greater than the critical value, we reject the null hypothesis and conclude that there is significant evidence that the proportion of women is different from 0.5.

3.

Z value for 95% confidence interval is 1.96

95% confidence interval is,

(0.3 - 1.96 * 0.0913, 0.3 + 1.96 * 0.0913)

= (0.121052, 0.478948)

4.

The preceding results suggests that the the proportion of women is under 0.5 and the sequence is also not random, so, the sample is biased against women. No, the the sample was not obtained in a random sequence. Yes, as the sequence are not random, we have problems with these results.