Question

A certain financial services company uses surveys of adults age 18 and older to determine if personal financial fitness is changing over time. Suppose that in February 2012, a sample of 1,000 adults showed 410 indicating that their financial security was more than fair. In February 2010, a sample of 1,100 adults showed 385 indicating that their financial security was more than fair.

(a)

State the hypotheses that can be used to test for a significant difference between the population proportions for the two years. (Let p₁ = population proportion saying financial security more than fair in 2012 and p₂ = population proportion saying financial security more than fair in 2010.)

H₀: p₁ − p₂ ≠ 0

H_a: p₁ − p₂ = 0

H₀: p₁ − p₂ = 0

H_a: p₁ − p₂ ≠ 0

    

H₀: p₁ − p₂ > 0

H_a: p₁ − p₂ ≤ 0

H₀: p₁ − p₂ ≤ 0

H_a: p₁ − p₂ > 0

H₀: p₁ − p₂ ≥ 0

H_a: p₁ − p₂ < 0

(b)

What is the sample proportion indicating that their financial security was more than fair in 2012?

What is the sample proportion indicating that their financial security was more than fair in 2010?

(c)

Conduct the hypothesis test and compute the p-value. At a 0.05 level of significance, what is your conclusion?

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to four decimal places.)

p-value =

State your conclusion.

Do not reject H₀. There is sufficient evidence to conclude the population proportions are not equal. The data suggest that there has been a change in the population proportion saying that their financial security is more than fair.Reject H₀. There is insufficient evidence to conclude the population proportions are not equal. The data do not suggest that there has been a change in the population proportion saying that their financial security is more than fair.     Do not reject H₀. There is insufficient evidence to conclude the population proportions are not equal. The data do not suggest that there has been a change in the population proportion saying that their financial security is more than fair.Reject H₀. There is sufficient evidence to conclude the population proportions are not equal. The data suggest that there has been a change in the population proportion saying that their financial security is more than fair.

(d)

What is the 95% confidence interval estimate of the difference between the two population proportions? (Round your answers to four decimal places.)

to

Answer 1

To Test :-

H₀: p₁ − p₂ = 0

H_a: p₁ − p₂ ≠ 0

Test Statistic :-

is the pooled estimate of the proportion P
= ( x1 + x2) / ( n1 + n2)
= ( 410 + 385 ) / ( 1000 + 1100 )
= 0.3786

Z = 2.83

Test Criteria :-
Reject null hypothesis if

= 2.83 > 1.96, hence we reject the null hypothesis
Conclusion :- We Reject H0

Decision based on P value
P value = 2 * P ( Z > 2.83 )
P value = 0.0047
Reject null hypothesis if P value <
Since P value = 0.0047 < 0.05, hence we reject the null hypothesis
Conclusion :- We Reject H0

Reject H₀. There is sufficient evidence to conclude the population proportions are not equal. The data suggest that there has been a change in the population proportion saying that their financial security is more than fair.

n1 = 1000
n2 = 1100


Lower Limit =
upper Limit =
95% Confidence interval is ( 0.0185 , 0.1015 )

( 0.0185 < ( P1 - P2 ) < 0.1015 )