Question

Born between 1980 and \(1990,\) Generation Next have lived in a post-Cold War world and a time of relative economic prosperity in America, but they have also experienced September 11 th and the fear of another attack, two Gulf Wars, the tragedy at Columbine High School, Hurricane Katrina, and the increasing polarization of public discourse. More than any who came before, Generation Next is engaged with technology, and the vast majority is dependent upon it. \({ }^{15}\) Suppose that a survey of 500 female and 500 male students in Generation Next, 345 of the females and 365 of the males reported that they decided to attend college in order to make more money. a. Construct a \(98 \%\) confidence interval for the difference in the proportions of female and male students who decided to attend college in order to make more money. b. What does it mean to say that you are "98\% confident"? c. Based on the confidence interval in part a, can you conclude that there is a difference in the proportions of female and male students who decided to attend college in order to make more money?

Short answer

In summary, we calculated the sample proportions for female and male students who decided to attend college in order to make more money, found the standard error of the difference in proportions, and constructed a 98% confidence interval using the z-score. Our confidence interval was [-0.114, 0.034], which includes zero, indicating that we cannot conclude that there is a significant difference in the proportions of female and male students who attended college to make more money. Being 98% confident refers to our level of confidence in our method of estimation, implying that in 98 out of 100 trials, we would expect the true population difference in proportions to lie within the calculated interval.

Step-by-step solution

Calculate the sample proportions for females (\(\hat{p}_1\)) and males (\(\hat{p}_2\))

The number of female students surveyed is \(n_1 = 500\), and 345 of them decided to attend college to make more money. Therefore, the sample proportion for females is \(\hat{p}_1 = \frac{345}{500}\) The number of male students surveyed is \(n_2 = 500\), and 365 of them decided to attend college to make more money. Therefore, the sample proportion for males is \(\hat{p}_2 = \frac{365}{500}\) Calculate the sample proportions: \(\hat{p}_1 = \frac{345}{500} = 0.69\) \(\hat{p}_2 = \frac{365}{500} = 0.73\)

Calculate the standard error of the difference in proportions

The standard error of the difference in proportions is given by the formula: \(SE = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}\) Using the values of \(\hat{p}_1\), \(\hat{p}_2\), \(n_1\), and \(n_2\), we can find the standard error: \(SE = \sqrt{\frac{0.69(1 - 0.69)}{500} + \frac{0.73(1 - 0.73)}{500}} \approx 0.0317\)

Calculate the 98% confidence interval for the difference in proportions

To construct the 98% confidence interval, we will use the following formula: \((\hat{p}_1 - \hat{p}_2) \pm z * SE\) Where \(z\) is the z-score corresponding to the desired level of confidence (98%). For a 98% confidence interval, the z-score is approximately \(2.33\). Now, we can calculate the confidence interval: \((0.69 - 0.73) \pm 2.33 * 0.0317\) \((-0.04) \pm 2.33 * 0.0317\) \([-0.114, 0.034]\)

Interpret the confidence interval and answer the questions

a) The 98% confidence interval for the difference in the proportions of female and male students who decided to attend college in order to make more money is \([-0.114, 0.034]\). b) Saying we are 98% confident means that if we were to conduct the same survey multiple times, we would expect the true population difference in proportions to lie within the calculated interval in 98 out of 100 trials. It's important to note that this doesn't mean there's a 98% chance that the true proportion is within the interval; rather, it's a statement about our level of confidence in our method of estimation. c) Since the confidence interval includes zero (\(0\)), we cannot conclude that there is a significant difference in the proportions of female and male students who decided to attend college in order to make more money. If the confidence interval did not include zero, we would be able to conclude that there is a significant difference between the proportions.

Key concepts

Understanding Sample Proportions

Sample proportions are used to estimate the true proportion of a population with a certain characteristic. In the case of the exercise, sample proportions refer to the proportion of female and male students within Generation Next who chose to attend college to make more money. The exercise involves two groups with sample sizes of 500 each, which are then used to calculate the individual proportions. For example, a sample proportion of 0.69 for females means that 69% of the surveyed female students attended college to increase their earnings.
These proportions are key when we intend to make inferences about the wider population of Generation Next students. These inferences can then form the basis for understanding broader social trends or assessing the effectiveness of certain policies or viewpoints that influenced their decision to pursue higher education.

Demystifying Standard Error

The concept of standard error is critical when estimating population parameters from sample data. It measures the variability or precision of a sample statistic such as a sample proportion. For instance, in our textbook solution, the standard error helps us understand the variability of the difference in proportions between female and male students. Simply put, a smaller standard error suggests that the sample proportion is likely to be closer to the true population proportion.
By calculating the standard error, which in the provided exercise is approximately 0.0317, we get a sense of how much sampling fluctuation we might expect. This serves as a foundation for constructing confidence intervals, enabling us to offer an educated guess about the true difference in proportions with an attached level of certainty.

Z-score: A Bridge Between Sample and Population

The z-score plays a crucial role when we transition from a sample statistic to making inferences about the entire population. It represents the number of standard deviations an observation is from the mean. As part of confidence interval calculations, the z-score corresponds to the desired level of confidence. In the exercise, a z-score of about 2.33 tells us that we are looking at a point 2.33 standard deviations away from the mean on the standard normal distribution, which is pivotal for constructing a 98% confidence interval.
When using the z-score to create confidence intervals, we essentially say, 'Given what we know from our sample, we believe the true population parameter lies within this range with a certain confidence level.' It's a way to quantify uncertainty in our estimates.

Statistical Significance and Confidence Intervals

The term 'statistical significance' often confuses students, but it's essential for testing hypotheses. It relates to the likelihood that the observed difference between groups (such as our female and male students) is due to chance rather than a true difference in the population. If a confidence interval does not contain the null value (often zero for difference in proportions), we consider the difference statistically significant.
In our example, since the 98% confidence interval includes zero (-0.114 to 0.034), we cannot claim a significant difference in proportions. Statistical significance is the cornerstone of hypothesis testing; it helps us make informed conclusions about whether there's enough evidence to support a particular claim about our population.