Question

1. Would you favor spending more federal tax money on the arts? Of a random sample of n₁ = 90 politically conservative voters, r₁ = 20 responded yes. Another random sample of n₂ = 84 politically moderate voters showed that r₂ = 22responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use α = 0.05.

(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 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. The number of trials is sufficiently large.

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

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 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

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

Reject the null hypothesis, there is sufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.

Fail to reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.     

Reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.

Fail to reject the null hypothesis, there is sufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.

Answer 1

Part a)

α = 0.05

To Test :-

H0 :- P1 = P2
H1 :- P1 < P2

part b)

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

p̂1 = 20 / 90 = 0.2222
p̂2 = 22 / 84 = 0.2619

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 = ( 20 + 22 ) / ( 90 + 84 )
p̂ = 0.2414
q̂ = 1 - p̂ = 0.7586
Z = ( 0.2222 - 0.2619) / √( 0.2414 * 0.7586 * (1/90 + 1/84) )
Z = -0.61

part c)

P value = P ( Z < -0.61 ) = 0.2705

part d)

Reject null hypothesis if P value < α = 0.05
Since P value = 0.2705 > 0.05, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

At the α = 0.05 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 conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.