Question

A random sample of n₁ = 283 voters registered in the state of California showed that 140 voted in the last general election. A random sample of n₂ = 217 registered voters in the state of Colorado showed that 129 voted in the most recent general election. Do these data indicate that the population proportion of voter turnout in Colorado is higher than that in California? Use a 5% level of significance.

(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 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.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.)


(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 not 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 statistically significant.

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

Fail to reject the null hypothesis, there is sufficient evidence that the proportion of voter turnout in Colorado is greater than that in California.Fail to reject the null hypothesis, there is insufficient evidence that the proportion of voter turnout in Colorado is greater than that in California.    Reject the null hypothesis, there is sufficient evidence that the proportion of voter turnout in Colorado is greater than that in California.Reject the null hypothesis, there is insufficient evidence that the proportion of voter turnout in Colorado is greater than that in California.

Answer 1

Given : n1=283 , X1=140 , n2=217 , X2=129

The sample proportions are ,

The pooled estimate is ,

Q=1-P=0.4620

Let , be the population proportion for the voters registered in the state of California be the population proportion for the voters registered in the state of Colorado

(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=0.0132 <

Therefore , reject Ho

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

(f) Conclusion : Reject the null hypothesis, there is sufficient evidence that the proportion of voter turnout in Colorado is greater than that in California.