Question

Based on information from a previous study, r = 34 people out of a random sample of n = 101 adult Americans who did not attend college believe in extraterrestrials. However, out of a random sample of n2 = 101 adult Americans who did attend college, '2 = 47 claim that they believe in extraterrestrials. Does this indicate that the proportion of people who attended college and who believe in extraterrestrials is higher than the proportion who did not attend college? Use a = 0.01 (a) What is the level of significance? State the null and alternate hypotheses. OHO: P1 = Pz; H:P, <P o Ho: P, <P₂i Hq : P₂ = P2 Ho: P1 = P2; 47: P. # P2 O Ho: P1 = P2; H:P > D2 (b) What sampling distribution will you use? What assumptions are you making? The standard normal. The number of trials is sufficiently large. The Student's t. The number of trials is sufficiently large. The Student's t. We assume the population distributions are approximately normal. The standard normal. We assume the population distributions are approximately normal. What is the value of the sample test statistic? (Test the difference 2, P. Do not use rounded values. Round your final answer to two decimal places.) (c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

Answer 1

a)

alpha = 0.01

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p1​=p2​

Ha: p1​>p2​

b)

The standard normal. The number of trials is sufficiently large.

The standard normal. we assume the population distribution are approximately normal.

For sample 1, we have that the sample size is N1 = 101, the number of favorable cases is X1 = 34, so then the sample proportion is 1 0.3366 For sample 2, we have that the sample size is N2 = 101, the number of favorable cases is X2 = 47, so then the sample proportion is f2 = = 161 = 0.4653 X1+X2_ The value of the pooled proportion is computed as p= 1+* Ni+N2 34+47 +0.401 101+101 Also, the given significance level is a = 0.05. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: P1 = P2 Ha : P1 <P2 This corresponds to a left-tailed test, for which a z-test for two population proportions needs to be conducted. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a left- tailed test is ze = -1.64. The rejection region for this left-tailed test is R= {z:z< -1.64} (3) Test Statistics The z-statistic is computed as follows:

Z = P1 – P2 VẼ(1-0)(1/ +1/02) * 0.3366 – 0.4653 = = -0.401 (1 – 0.401)(1/101 +1/101) -1.866 (4) Decision about the null hypothesis Since it is observed that z = -1.866<zc = -1.64, it is then concluded that the null hypothesis is rejected. Using the P-value approach: The p-value is p = 0.031, and since p = 0.031 < 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 proportion pı is less than P2, at the 0.05 significance level.

test statistic z =−1.866

The p-value is p = 0.031

$normaldistributiongrapher.php?mean=0&sig$

d)

At alpha = 0.01 level we reject the null hypothesis and conclude the data are not statistically significant.

e)

Reject the null hypothesis, there is sufficient evidence that the proportion of adults that attended college who believe in extraterrestrial is higher than that of adults who did not attend college.