Question

Country Financial, a financial services company, uses surveys of adults age and older to determine if personal financial fitness is changing over time (USA Today, April 4, 2012). In February of 2012, a sample of adults showed indicating that their financial security was more than fair. In February of 2010, a sample of adults showed 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.
- Select your answer -greater than 0greater than or equal to 0less than 0less than or equal to 0equal to 0not equal to 0Item 1
- Select your answer -greater than 0greater than or equal to 0less than 0less than or equal to 0equal to 0not equal to 0Item 2
b. What is the sample proportion indicating that their financial security was more than fair in 2012? Round your answer to two decimal places.
In 2010? Round your answer to two decimal places.
c. Conduct the hypothesis test and compute the -value. Round your answer to four decimal places.
-value =
At level of significance, what is your conclusion?
The input in the box below will not be graded, but may be reviewed and considered by your instructor.
d. What is the confidence interval estimate of the difference between the two population proportions? Round your answers to four decimal places.
Confidence Interval ( to )
e. Based on your conclusion of the previous question, can you say the population proportion of adults saying that their financial security is more than fair has increased or decreased?

Answer 1

Answer:

Step-by-step explanation:

Hello!

The variables of interest are:

X₁: Number of adults age 18 and older in 2012whose personal financial security is fair. in a sample of 1000.

X₁ ~Bi(n₁;p₁)

n₁= 1000

x₁= 410

p'₁= x₁/n₁= 410/1000= 0.41

X₂: Number of adults age 18 and older in 2010 whose personal financial security is fair. in a sample of 900.

X₂ ~Bi(n₂;p₂)

n₂= 900

x₂= 315

p'₂= x₂/n₂= 315/900= 0.35

In February 2012, a sample of 1000 adults showed 410 indicating that their financial security was more than fair. In Feb 2010, a sample of 900 adults showed 315 indicating that their financial security was more than fair.

a.

If the objective is to test whether financial fitness is changing over time, you have to make a two-tailed test:

H₀: p₁ - p₂ = 0

H₁: p₁ - p₂ ≠ 0

b.

To calculate the sample proportion you have to divide the number of success "x" by the sample size:

Sample proportion for 2012

p'₁= x₁/n₁= 410/1000= 0.41

Sample proportion for 2010

p'₂= x₂/n₂= 315/900= 0.35

c.

The test statistic is:

$Z=\frac{(p'_1-p'_2)-(p_1-p_2)}{p'(1-p')\sqrt{\frac{1}{n_1}+\frac{1}{n_2} } } = \frac{(0.41-0.35)-0}{\sqrt{0.38*0.62*(\frac{1}{1000} \frac{1}{900} }) } = 2.69$

$p'= \frac{x_1+x_2}{n_1+n_2}= \frac{410+315}{1000+900}= 0.38$

The p-value is 0.00714

α: 0.05

The p-value is less than α, so the decision is to reject the null hypothesis.

d.

$p'_1-p'_2= 0.41-0.35= 0.06$

$Z_{0.975}= 1.965$

[$(p'_1-p'_2)$± $Z_{1-\alpha /2}$*$\sqrt{p'(1-p')(\frac{1}{n_1}+\frac{1}{n_2}) } }$]

[0.06±(1.965*0.022)]

[0.01677; 0.10323]

e.

In item c. the decision was to not reject the null hypothesis. This means at a significance level of 5% that the population proportions of the financial fitness of adults age 18 and older in the years 2012 and 2010 are the same.

I hope it helps!