Question

Would you favor spending more federal tax money on the arts? Of a random sample of n₁ = 95 politically conservative voters, r₁ = 16 responded yes. Another random sample of n₂ = 77 politically moderate voters showed that r₂ = 20 responded 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 > p₂H₀: p₁ < p₂; H₁: p₁ = p₂    H₀: p₁ = p₂; H₁: p₁ < p₂H₀: p₁ = p₂; H₁: p₁ ≠ p₂

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


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

(e) 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 fail to reject the null hypothesis and conclude the data are 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 reject the null hypothesis and conclude the data are not statistically significant.

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

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.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.    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 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

Given : n1=95 , r1=X1=16 , n2=77 , r2=X2=20

The sample proportions are ,

The pooled estimate is ,

Q=1-P=0.7907

Let , be the population proportion for the politically conservative voters responded yes be the population proportion for the politically moderate voters responded yes.

(a) The level of signifiacnce is

Hypothesis : Vs  

(b) The sampling distribution is standard normal

We assume that the population distribution are approximately normal.

The value of the sample test statistic is ,

(c) The p-value is ,

p-value=

; From standard normal distribution table

(d)

(e) Decision : Here , p-value=0.0721 >

Therefore , fail to reject Ho

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

(f) Conclusion : 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