Question

Based on information from a previous study, r₁ = 39 people out of a random sample of n₁ = 102 adult Americans who did not attend college believe in extraterrestrials. However, out of a random sample of n₂ = 102 adult Americans who did attend college, r₂ = 45 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 α = 0.01.

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀: p₁ = p₂; H₁: p₁ > p₂

H₀: p₁ < p₂; H₁: p₁ = p₂    

H₀: p₁ = p₂; H₁: p₁ ≠ p₂

H₀: p₁ = p₂; H₁: p₁ < p₂

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume the population distributions are approximately normal.

The Student's t. We assume the population distributions are approximately normal.    

The Student's t. The number of trials is sufficiently large.

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

What is the value of the sample test statistic? (Test the difference p₁ − p₂. 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.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.    

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

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that the proportion of adults that attended college who believe in extraterrestrials is higher than that of adults who did not attend college.

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

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

Fail to reject the null hypothesis, there is insufficient evidence that the proportion of adults that attended college who believe in extraterrestrials is higher than that of adults who did not attend college.

Answer 1

Part a)

α = 0.01

H0: p1 = p2; H1: p1 < p2

Part b)

The standard normal. We assume the population distributions are approximately normal.

p̂1 = 39 / 102 = 0.3824
p̂2 = 45 / 102 = 0.4412

Test Statistic :-
Z = ( p̂1 - p̂2 ) / √(p̂ * q̂ * (1/n1 + 1/n2) ) )
p̂ is the pooled estimate of the proportion P
p̂ = ( x1 + x2) / ( n1 + n2)
p̂ = ( 39 + 45 ) / ( 102 + 102 )
p̂ = 0.4118
q̂ = 1 - p̂ = 0.5882
Z = ( 0.3824 - 0.4412) / √( 0.4118 * 0.5882 * (1/102 + 1/102) )
Z = -0.85

Part c)

P value = P ( Z < -0.8536 ) = 0.1976

Reject null hypothesis if P value < α = 0.01
Since P value = 0.1976 > 0.01, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Part d)

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.    

Part e)

Fail to reject the null hypothesis, there is insufficient evidence that the proportion of adults that attended college who believe in extraterrestrials is higher than that of adults who did not attend college.