Question

In a survey of 1000 women age 22 to 35 who work full time, 540 indicated that they would be willing to give up some personal time in order to make more money (USA Today. March 4, 2010 ). The sample was selected in a way that was designed to produce a sample that was representative of women in the targeted age group. a. Do the sample data provide convincing evidence that the majority of women age 22 to 35 who work full-time would be willing to give up some personal time for more money? Test the relevant hypotheses using \(\alpha=.01\). b. Would it be reasonable to generalize the conclusion from Part (a) to all working women? Explain why or why not.

Short answer

a) If our calculated Z-value from Step 2 is larger than the critical value of 2.33, we can conclude that the majority of women age 22 to 35 who work full-time would be willing to give up some personal time for more money. b) No, the conclusion cannot be generalized to all working women due to the specific age and work status of women in the study.

Step-by-step solution

Set up the hypotheses

We can express our hypotheses as follows: Null hypothesis (H0): p = 0.5 - The proportion of women who would give up personal time is the same as those who would not. Alternative hypothesis (Ha): p > 0.5 - The proportion of women who would give up personal time is more than those who would not.

Calculate the test statistic

The Z-value can be calculated using the following formula: Z = \(( \hat{p} - p0) / \sqrt{ \frac{p0*(1-p0)}{n}} \) Where, \hat{p} = proportion in the sample = 540/1000 = 0.54 p0 = assumed population proportion under H0 = 0.5 n = sample size = 1000 The Z-value is calculated as: Z = \( (0.54 - 0.5) / \sqrt{(0.5*(1-0.5))/1000} \)

Find the critical value for \(\alpha =.01\)

The critical value for a one-tailed test at \(\alpha = .01\) is Z = 2.33. If our calculated Z-value is greater than this, we reject the null hypothesis.

Compare the test statistic to the critical value and make a decision

If the calculated Z-value from Step 2 is larger than 2.33, we reject the null hypothesis and conclude that the majority of women age 22 to 35 who work full-time would give up some personal time for more money.

Answer Question b

If the sample was truly representative of the population, we can infer the same for the entire population. However, we cannot generalize this conclusion to all working women, as the sample was specific to women between ages 22 and 35 who work full-time.

Key concepts

Null Hypothesis

In hypothesis testing, the null hypothesis is a foundational concept. It is often denoted as \( H_0 \). This hypothesis represents a statement of no effect or no difference. In other words, it suggests that any observed difference in a sample is due to random chance. For the problem at hand, the null hypothesis is that the proportion of women willing to give up personal time for money is the same as those who aren't. Formally, it's expressed as \( p = 0.5 \). This means that 50% of women would and 50% wouldn't give up personal time.

Alternative Hypothesis

Contrary to the null hypothesis, the alternative hypothesis is denoted by \( H_a \). It reflects the researcher's claim or what they aim to prove. Here, it suggests a significant effect or difference, generally contradicting the null hypothesis. For our exercise, the alternative hypothesis states that more than half of women in the specified group are willing to sacrifice personal time for extra money, not just an even split. Mathematically, this means \( p > 0.5 \). If evidence supports this hypothesis, then the claim of majority holds true.

Z-value Calculation

The Z-value, also called the Z-score, is a statistical measure that shows how far away a data point is from the mean in terms of standard deviations. It's critical in determining whether to reject the null hypothesis. In the formula \( Z = \frac{(\hat{p} - p_0)}{\sqrt{\frac{p_0 (1-p_0)}{n}}} \):

• \( \hat{p} \) is the sample proportion.
• \( p_0 \) is the assumed population proportion under the null hypothesis.
• \( n \) refers to the sample size.

In this problem, while \( \hat{p} = 0.54 \) and \( p_0 = 0.5 \), with \( n \) of 1000, this calculation helps us decide whether the evidence strongly contradicts \( H_0 \). If the Z-value is greater than the critical value of 2.33 (for a one-tailed test with \( \alpha = 0.01 \)), we reject \( H_0 \).

Population Proportion

The population proportion \( p \) represents the fraction of the total population that demonstrates a particular attribute. In hypothesis testing, it's an assumed value based on prior knowledge or expectations, under the null hypothesis. In this exercise, the population proportion under \( H_0 \) is set at 0.5. This proportion serves as a benchmark to test against our observed data from the sample. Understanding this concept helps frame our expectations and guides the null hypothesis formation.

Sample Size

Sample size, designated as \( n \), is a crucial part of hypothesis testing. It refers to the number of observations or data points collected from the population. A sufficiently large sample size is important to ensure the results are reliable and not due to random variation. In our exercise, the sample size is 1000, indicating a substantial group, giving weight to the findings and making them more credible. The larger the sample size, the more likely it is to accurately represent the larger population, lending confidence to the conclusions drawn from the test.