Question

News article reports that "Americans have differing views on two potentially inconvenient and invasive practices that airports could implement to uncover potential terrorist attacks." This news piece was based on a survey conducted among a random sample of 1116 adults nationwide, interviewed by telephone November 7-10, 2010, where one of the questions on the survey was "Some airports are now using 'full-body' digital x-ray machines to electronically screen passengers in airport security lines. Do you think these new x-ray machines should or should not be used at airports?" Below is a summary of responses based on party affiliation. Round all numeric answers to 4 decimal places.
Republican Democrat Independent Total
Should 258 294 345 897
Should not 37 54 75 166
Don't know/No answer 16 15 22 53
Total 311 363 442 1116

1. What is the proportion of Republicans who think the full-body scans should be applied in airports?

2. What is the proportion of Democrats who think the full-body scans should be applied in airports?

3. What are the correct hypotheses for a hypothesis test evaluating whether there is a difference in the proportion of Republicans and Democrats who think the full-body scans should be applied in airports? Assume that all relevant conditions are met.

A. H0:p1?p2=0; HA:p1?p2>0
B. H0:p1?p2=0 ; HA:p1?p2?0
C. H0:p1?p2=0 ; HA:p1?p2<0

4. Calculate the pooled estimate of a proportion for this test.

5. Calculate the standard error for this test.

6. Calculate the test statistic for this hypothesis test.

7. Calculate the p-value for this hypothesis test.

8. What is your conclusion using alpha = .01

A. Do not reject H0
B. Reject H0

Answer 1

Answer:

1) p1=0.83

2) p2=0.81

3) Option B

4) p=0.82

5) s=0.03

6) z=0.67

7) P=0.50

8) Do not reject H0

Step-by-step explanation:

1) The proportion of Republicans who think the full-body scans should be applied in airports is equal to the ratio between the republicans that think it should over the total Republicans in the poll:

$p=X/n=258/311=0.83$

2) The proportion of Democrats who think the full-body scans should be applied in airports is equal to:

$p=X/n=294/363=0.81$

3) As we want to know if there is a difference in the proportions calculated, not if one is higher or lower than the other, the correct option is B:

$H_0:p_1-p_2=0\\\\H_a:p_1-p_2\neq0$

4) The pooled estimate of a proportion is the average of both proportions:

$\bar{p}=\frac{p_1+p_2}{2}= \frac{0.83+0.81}{2}=0.82$

5) The standard error can be calculated as:

$s=\sqrt{\frac{\bar{p}(1-\bar{p})}{n_1} +\frac{\bar{p}(1-\bar{p})}{n_2}} \\\\s=\sqrt{\frac{0.82*0.18}{311} +\frac{0.82*0.18}{363}} \\\\s=\sqrt{0.00047+0.00041} =\sqrt{0.00088}=0.03$

6) The z-statistic can be calculated as:

$z=\frac{p_1-p_2}{s}=\frac{0.83-0.81}{0.03}=\frac{0.02}{0.03}=0.67$

7) The P-value for z=0.67 is P=0.50 (from the standard normal distribution table).

$2P(Z>|z|)=0.50286$

8) If the significance level is 0.01, the P-value is bigger than the significance level. The effect is not significanct. The null hypothesis is not rejected.

There is not enough evidence to say that both proportions are different.

A. Do not reject H0