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A study asked 1,924 male and 3,666 female undergraduate college students their favorite color. A \(95 \%\) confidence interval for the difference between the proportions of males and females whose favorite color is black \(\left(p_{\text {male }}-p_{\text {female }}\right)\) was calculated to be (0.02,0.06) . Based on this information, determine if the following statements are true or false, and explain your reasoning for each statement you identify as false. \(^{28}\) (a) We are \(95 \%\) confident that the true proportion of males whose favorite color is black is \(2 \%\) lower to \(6 \%\) higher than the true proportion of females whose favorite color is black. (b) We are \(95 \%\) confident that the true proportion of males whose favorite color is black is \(2 \%\) to \(6 \%\) higher than the true proportion of females whose favorite color is black. (c) \(95 \%\) of random samples will produce \(95 \%\) confidence intervals that include the true difference between the population proportions of males and females whose favorite color is black. (d) We can conclude that there is a significant difference between the proportions of males and females whose favorite color is black and that the difference between the two sample proportions is too large to plausibly be due to chance. (e) The \(95 \%\) confidence interval for \(\left(p_{\text {female }}-p_{\text {male }}\right)\) cannot be calculated with only the information given in this exercise.

Step by step solution

01

Understanding the Confidence Interval

We are given a 95% confidence interval for the difference between the proportions of males and females whose favorite color is black: (0.02, 0.06). This interval estimates the difference in proportions as being between 2% to 6%, suggesting that the male proportion is higher.

02

Analyzing Statement (a)

Statement (a) is false because the interval is from 0.02 to 0.06, suggesting the male proportion is higher. However, the statement mentions '2% lower to 6% higher', indicating a misunderstanding. The correct interpretation is the male proportion is between 2% and 6% higher.

03

Analyzing Statement (b)

Statement (b) is true. The interval (0.02, 0.06) means that the proportion of males whose favorite color is black is 2% to 6% higher than that of females.

04

Analyzing Statement (c)

Statement (c) is true. By definition, 95% confidence means that 95% of such calculated intervals would contain the true difference in population proportions if we were to take many random samples.

05

Analyzing Statement (d)

Statement (d) is true. The fact that the entire confidence interval lies above zero (0.02, 0.06) indicates that there is a statistically significant difference between the two proportions.

06

Analyzing Statement (e)

Statement (e) is false. The confidence interval for \(p_{\text{female}} - p_{\text{male}}\) is simply the negative of the given interval, i.e., (-0.06, -0.02), obtainable from the provided information.

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Key Concepts

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

Difference in Proportions

One of the core concepts in this exercise is understanding the difference in proportions. When we talk about the difference in proportions, we are comparing two group percentages — in this case, males and females whose favorite color is black.

• The difference tells us how much larger or smaller one group’s proportion is compared to the other’s.
• A positive difference, such as in this exercise's interval (0.02, 0.06), indicates the first group (males) has a higher proportion.
• A negative difference would indicate the opposite.

This understanding helps in interpreting results from studies and forming conclusions. When the entire interval lies above zero, as it does here, we are confident that there is a real difference favoring the male proportion in this context.

Statistical Significance

Statistical significance is about how certain we are that an observed effect or measure isn't due to random chance. Here, a 95% confidence interval for proportions is used to assess significance.

• This interval suggests that we are 95% confident in the range it provides for the difference.
• The fact that the interval (0.02, 0.06) does not cross zero indicates a statistically significant difference.
• If zero were inside the interval, it would suggest no significant difference might exist because zero implies equal proportions.

The statistical significance allows us to conclude that males and females have differing favorite color preferences.

Sample Proportions

Sample proportions are estimates based on survey data from a subset of the population. They are central to understanding many real-world phenomena, including how groups differ in characteristics or preferences.

• In this exercise, we have sample data from 1,924 males and 3,666 females.
• The sample proportion of a trait (favorite color) is calculated by dividing the number who have the trait by the total surveyed in the sample group.
• These proportions help us estimate the difference between the larger population proportions, leading to the confidence interval calculation.

By comparing these, we can draw inferences about similar populations beyond the sample.

Interpretation in Statistics

Interpretation in statistics involves making sense of the numbers to answer a specific question or hypothesis. It converts raw data into meaningful insights.

• The exercise uses the confidence interval to interpret the difference between male and female preferences.
• It's essential to read such intervals correctly: (0.02, 0.06) suggests 2% to 6% more males prefer black than females.
• Misinterpretations can easily occur if the direction and range of the intervals are not correctly understood, leading to false claims.

Being able to accurately interpret data is crucial in forming reliable conclusions and avoiding errors in understanding the statistics presented.