Question

In February 2008, an organization surveyed 1040 adults aged 18 or older and found that 541 believed they would not have enough money to live comfortably in retirement. Does the sample evidence suggest that the majority of adults believe they will have enough money in retirement?

a) At the .05 level of significance, does the sample evidence suggest that the majority of adults believe they will have enough money to live comfortably in retirement? Use the classical approach.

b) Based on your results of the hypothesis test—the decision you made in part a, what type of error might have occurred? Please explain.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P < 0.50
Alternative hypothesis: P > 0.50

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected only if the sample proportion is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (σ) and compute the z-score test statistic (z).

σ = sqrt[ P * ( 1 - P ) / n ]

σ = 0.015504
z = (p - P) / σ

z = 1.30

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is more than 1.30. Thus, the P-value = 0.0968.

Interpret results. Since the P-value (0.0968) is more than the significance level (0.05), we have to accept the null hypothesis.

b) As we have accepted the null hypothesis, Type II error might have occured.