91Ó°ÊÓ

Wage discrimination? According to a union agreement, the mean income for all senior-level assembly-line workers in a large company equals $$\$ 500$$ per week. A representative of a women's group decides to analyze whether the mean income for female employees matches this norm. For a random sample of nine female employees, using software she obtains a \(95 \%\) confidence interval of (371,509) . Explain what is wrong with each of the following interpretations of this interval. a. We infer that \(95 \%\) of the women in the population have income between $$\$ 371$$ and $$\$ 509$$ per week. b. If random samples of nine women were repeatedly selected, then \(95 \%\) of the time the sample mean income would be between \(\$ 371\) and \(\$ 509\). c. We can be \(95 \%\) confident that \(\bar{x}\) is between \(\$ 371\) and \(\$ 509\). d. If we repeatedly sampled the entire population, then \(95 \%\) of the time the population mean would be between $$\$ 371$$ and $$\$ 509$$

Step by step solution

01

Understanding Confidence Intervals

A confidence interval provides a range of values that likely includes a population parameter, such as the mean, based on a sample from a population. A 95% confidence interval means we are 95% confident that the true population mean falls within this range.

02

Analyzing Interpretation a

Statement a wrongly interprets the confidence interval as pertaining to individual women's incomes. A confidence interval is about where the population mean is likely to fall, not the range of individual incomes.

03

Analyzing Interpretation b

Statement b suggests that the sample mean itself will fall within the confidence interval 95% of the time when sampling, which is incorrect. The interval is about where the population mean is likely to be, not where repeated sample means will lie.

04

Analyzing Interpretation c

Statement c incorrectly claims confidence about the sample mean \( \bar{x} \), which we already have. The confidence interval is about the range where we expect the population mean to be, based on the sample mean.

05

Analyzing Interpretation d

In statement d, the interpretation suggests that repeatedly sampling will result in a population mean within the interval, which is incorrect. The interval is not a predictor of where repeated calculated population means would fall, since the population mean is fixed.

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Key Concepts

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

Population Mean

The population mean is a core concept in statistics and refers to the average value of a characteristic for an entire population. It's a fixed value, though we often don't know it precisely due to practical limitations in collecting data. For example, in the context of wage discrimination, if we consider the income of all senior-level assembly-line workers, the population mean would be the average income for all these employees.

To estimate this population mean, statisticians rely on sampling since it is often impractical to measure every individual in the population. Although we strive to determine this mean, it's crucial to remember that the population mean itself doesn’t change. Instead, our estimates may vary depending on the sample we select.

• The population mean provides insights when examining potential discrepancies, such as wage differences among groups based on gender, experience, or other factors.
• It's a stable number that our confidence intervals aim to estimate more accurately.

Sample Mean

The sample mean, often symbolized as \( \bar{x} \), represents the average of data collected from a subset of a larger population. In studies where full data collection is unfeasible, like wage analysis of assembly-line workers, we rely heavily on the sample mean as an estimator for the population mean.

The sample mean gives us a snapshot of our sample group, helping infer broader population patterns. This estimate is pivotal because it forms the center of our confidence interval estimation.

• Calculating the sample mean involves summing all sample values and dividing by the number of observations \( (n) \).
• The sample mean is sensitive to the sample size—larger samples tend to provide a more accurate estimation of the population mean.

Understanding the difference between the sample mean and the population mean is crucial, particularly when interpreting results like confidence intervals.

Statistical Inference

Statistical inference is a process where we use data from a sample to make educated guesses about a population. This process relies heavily on probability and is foundational in determining estimates like the population mean.

When assessing the income levels of a specific group within a larger workforce, statistical inference allows us to make conclusions beyond the raw data of our sample. This approach isn't about individual values but the population parameters like the mean or proportion. A large portion of statistical inference involves creating confidence intervals and hypothesis testing.

• Confidence intervals help estimate the range where the population parameter might lie.
• Hypothesis testing can support or refute claims about population characteristics, such as wage equality across demographics.

Statistical inference equips analysts with the tools to navigate uncertainty and variability when making predictions based on sample data.

Confidence Level

The confidence level in statistics refers to the degree of certainty we have that our sample reflects the true population parameter. A common choice is a 95% confidence level, implying that we're 95% confident the true population mean lies within our calculated interval.

In the context of our example analyzing wages, a confidence interval with a 95% confidence level provides a range that likely contains the true mean income for the population of female workers. However, this does not mean 95% of all individuals will have incomes in this range.

• The confidence level primarily affects the width of the confidence interval—higher confidence levels widen the interval.
• Misinterpretations can occur if we confuse the confidence level with the probability of individual outcomes.

A solid grasp of the confidence level is essential in correctly interpreting statistical results and understanding their implications regarding population estimates.