Math  /  Data & Statistics

QuestionIn February 2008, an organization surveyed 1040 adults aged 18 and older and found that 535 believed they would not have enough money to live comfortably in retirement. Does the sample evidence suggest that a majority of adults in a certain country believe they will not have enough money in retirement? Use the α=0.1\alpha=0.1 level of significance.
Formula sheet What are the null and alternative hypotheses? H0\mathrm{H}_{0} : p \square \square versus H1\mathrm{H}_{1} : p \square \square Use technology to find the P -value. P -value == \square (Round to four decimal places as needed.) Choose the correct answer below. A. Since PP-value >α>\alpha, do not reject the null hypothesis and conclude that there is not sufficient evidence that a majority of adults in the United States believe they will not have enough money in retirement. B. Since PP-value <α<\alpha, do not reject the null hypothesis and conclude that there is sufficient evidence that a majority of adults in the United States believe they will not have enough money in retirement. C. Since PP-value >α>\alpha, reject the null hypothesis and conclude that there is not sufficient evidence that a majority of adults in the United States believe they will not have enough money in retirement. D. Since P -value <α<\alpha, reject the null hypothesis and conclude that there is sufficient evidence that a majority of adults in the United States believe they will not have enough money in retirement.

Studdy Solution

STEP 1

1. The sample size is 1040 adults.
2. 535 adults believe they will not have enough money to live comfortably in retirement.
3. We are testing at a significance level of α=0.1\alpha = 0.1.
4. We are conducting a hypothesis test for a proportion.

STEP 2

1. Define the null and alternative hypotheses.
2. Calculate the sample proportion.
3. Determine the test statistic.
4. Find the P-value.
5. Make a decision based on the P-value.

STEP 3

Define the null and alternative hypotheses:
H0:p0.5(The proportion is not a majority)\mathrm{H}_{0}: p \leq 0.5 \quad \text{(The proportion is not a majority)}
H1:p>0.5(The proportion is a majority)\mathrm{H}_{1}: p > 0.5 \quad \text{(The proportion is a majority)}

STEP 4

Calculate the sample proportion:
p^=5351040=0.5144\hat{p} = \frac{535}{1040} = 0.5144

STEP 5

Calculate the test statistic using the formula for a proportion:
z=p^0.50.5×(10.5)1040z = \frac{\hat{p} - 0.5}{\sqrt{\frac{0.5 \times (1 - 0.5)}{1040}}}
z=0.51440.50.2510401.16z = \frac{0.5144 - 0.5}{\sqrt{\frac{0.25}{1040}}} \approx 1.16

STEP 6

Use technology to find the P-value for z=1.16z = 1.16:
P-value 0.1230\approx 0.1230 (rounded to four decimal places)

STEP 7

Compare the P-value to α\alpha:
Since P-value =0.1230>α=0.1= 0.1230 > \alpha = 0.1, we do not reject the null hypothesis.
Choose the correct answer:
A. Since PP-value >α> \alpha, do not reject the null hypothesis and conclude that there is not sufficient evidence that a majority of adults in the United States believe they will not have enough money in retirement.

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