91Ó°ÊÓ

Employment by gender The study described in Exercise 10.16 also evaluated the weekly time spent in employment. This sample comprises men and women with a high level of labor force attachment. Software shows the results. \begin{tabular}{lcccc} Gender & \(\mathrm{N}\) & Mean & StDev & SE Mean \\ Men & 496 & 47.54 & 9.92 & 0.45 \\ Women & 476 & 42.01 & 6.53 & 0.30 \\ Difference \(=\mathrm{mu}(\mathrm{Men})-\mathrm{mu}(\) Women \()\) & \\ \hline \end{tabular} 95\% CI for difference: (4.477,6.583) T-Test of difference \(=0(\mathrm{vs} \neq):\) \(\mathrm{T}-\) Value \(=10.30\) p-value \(=0.000\) a. Does it seem plausible that employment has a normal distribution for each gender? Explain. b. What effect does the answer to part a have on inference comparing population means? What assumptions are made for the inferences in this table? c. Explain how to interpret the confidence interval. d. Refer to part c. Do you think that the population means are equal? Explain.

Step by step solution

01

Assess Normal Distribution Plausibility

To determine if employment has a normal distribution for each gender, we should consider factors such as sample size and standard deviation. With sample sizes of 496 for men and 476 for women, the Central Limit Theorem suggests that the distribution of sample means will approach normality regardless of the original distribution. However, without graphical evidence (like histograms) or tests for normality (such as the Shapiro-Wilk test), we can only assume normality based on sample size, not on distribution shape.

02

Consider the Effect of Normality on Inference

Inferences comparing population means rely on the assumption that data are normally distributed. If this assumption holds, the T-test and confidence intervals are valid. If not, the inferences might be inaccurate. However, due to the large sample sizes (n > 30), the inference is robust to deviations from normality.

03

Interpret the Confidence Interval

The 95% confidence interval for the difference in means between men and women is (4.477, 6.583). This means that we are 95% confident that the true difference in weekly employment hours between men and women falls between these values. The interval does not include zero, which suggests a statistically significant difference.

04

Evaluate Population Means Equality

The confidence interval for the difference does not contain zero, indicating that it is very unlikely that the true population means are equal. Additionally, the T-Test yields a p-value of 0.000, which strongly suggests that there is a significant difference in weekly employment hours between men and women.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval

A confidence interval is a statistical tool used to estimate the range in which a population parameter lies, based on sample data.
In the case of the employment study mentioned, the confidence interval for the difference in weekly working hours between men and women is given as (4.477, 6.583).
This means we are 95% confident that the true difference in average hours worked per week between men and women falls within this range.
What makes confidence intervals useful? Let's consider these points:

• **Confidence Level**: The percentage (95% in this case) indicates the probability that the interval contains the true parameter value.
• **Interval Bounds**: The interval not including zero suggests that men and women have different average working hours.
• **Sample Data Influence**: Larger sample sizes typically result in narrower confidence intervals, providing more precise estimates.

Confidence intervals provide valuable context, indicating not only the estimate of the difference but also the certainty of the estimate based on the data collected.
Explaining the failure to include zero in this interval, we can infer that there is a significant statistical difference in employment hours between these genders.

Normal Distribution

The normal distribution, commonly referred to as the "bell curve," is a continuous probability distribution that is symmetrical about the mean. It's foundational to many statistical tools and assumes that data are distributed in a pattern where most observations cluster around the mean.
In employment statistics, assuming a normal distribution allows us to make valid inferences about population means from sample data using the Central Limit Theorem.
The theorem states that with a large enough sample size, the distribution of the sample mean will approximate normality, even if the underlying distribution isn't normal. Key considerations include:

• **Large Sample Sizes**: The sample sizes for both men (496) and women (476) are sufficiently large (greater than 30) to rely on the Central Limit Theorem.
• **Robustness to Deviations**: Even if the employment data aren't perfectly normal, large sample sizes render the t-tests and confidence intervals robust to such deviations.
• **Empirical Validation**: Graphical representations like histograms or normality tests can further assert distribution assumptions.

Normal distribution assumptions are critical because they underpin the validity of many statistical methods employed to compare population parameters.

T-Test

The T-test is a statistical method used to determine whether there is a significant difference between the means of two groups. In the context of employment statistics, the T-test helps compare the average weekly working hours of men and women. Here's why the T-test is important in this study:

• **Hypothesis Testing**: We test whether the difference between men's and women's working hours is zero, which would mean the averages are equal (null hypothesis).
• **T-Value and p-value**: A high T-value (10.30 in this case) and a low p-value (0.000) suggest a significant difference between the two groups.
• **Assumptions**: The T-test assumes that the data is continuous, normally distributed (or approximately so), and that the variances are homogenous or similar between groups.

The results, with a p-value of 0.000, strongly reject the null hypothesis, indicating a significant difference in employment hours. Thus, using the T-test in this context provides a rigorous way to support claims about gender differences in work hours.