Question

The Chronicle of Higher Education (January 13, 1993) reported that \(72.1 \%\) of those responding to a national survey of college freshmen were attending the college of their first choice. Suppose that \(n=500\) students responded to the survey (the actual sample size was much larger). a. Using the sample size \(n=500\), calculate a \(99 \%\) confidence interval for the proportion of college students who are attending their first choice of college. b. Compute and interpret a \(95 \%\) confidence interval for the proportion of students who are not attending their first choice of college. c. The actual sample size for this survey was much larger than 500 . Would a confidence interval based on the actual sample size have been narrower or wider than the one computed in Part (a)?

Short answer

a. The 99% confidence interval for students attending their first choice of college is from 0.692 to 0.750. b. The 95% confidence interval for students not attending their first choice of college is from 0.245 to 0.313. c. A confidence interval based on the actual sample size would have been narrower than the one computed in part (a).

Step-by-step solution

Calculate 99% confidence interval for first choice

The formula for confidence interval is \(p \pm Z_{\frac{a}{2}} \cdot \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is proportion, \(n\) is the sample size and \(Z_{\frac{a}{2}}\) is the respective Z-score for a given confidence level. For 99% confidence, \(Z_{\frac{a}{2}} = 2.576\). Substitute \(p = 0.721\) and \(n = 500\) into the formula to receive the confidence interval.

Calculate 95% confidence interval for not first choice

The proportion of students not attending their first choice of college is \(1 - p = 1 - 0.721 = 0.279\). For 95% confidence, \(Z_{\frac{a}{2}} = 1.96\). Substitute these values into the formula from Step 1 to obtain the confidence interval.

Compare the width of confidence intervals

A confidence interval becomes narrower with a larger sample size. In part (c) it specifies the actual sample size was much larger than 500. An interval based on a larger sample size than 500, therefore, would have been narrower than the one computed in part (a).

Key concepts

Statistics

Statistics is the science of collecting, analyzing, and interpreting data to make informed decisions. It uses a combination of mathematics, probability, and computational techniques to convert raw data into useful information. In the context of our exercise, statistics helps us understand aspects of college freshmen's choices regarding their education through the construction of confidence intervals, which offer a likely range of values based on their responses to a survey.

For students grappling with statistical concepts, remember that every dataset provides a story about a particular population. In this case, it's about college freshmen and their college preferences. By delving into the statistics, they can not only get numerical results but can also paint a broader picture of the population's behavior or preferences, leading to valuable insights.

Data Analysis

Data analysis involves the systematic application of statistical and logical techniques to describe and illustrate, condense and recap, and evaluate data. The primary purpose is to extract useful information and develop conclusions. In our exercise, we examine survey data to better understand a trend within the student population. Through the calculated confidence intervals, we can analyze how confident we can be in our estimates of the entire college freshmen population's preferences.

Through a relatable example, such as students attending their first choice of college, data analysis becomes less daunting. It's akin to surveying a handful of friends about their favorite ice cream flavors to predict the next flavor the group might enjoy.

Confidence Level

The confidence level represents the degree of certainty that the calculated confidence interval includes the true population parameter. For example, a 99% confidence level means that if we were to take 100 different random samples and compute a confidence interval for each sample, we would expect 99 of the confidence intervals to contain the true population proportion.

A higher confidence level indicates a larger margin of error and a broader confidence interval. It's important to note that while a higher confidence level may seem like it would yield better results, it also means the interval will be less precise. This is the trade-off when choosing a confidence level.

Sample Size

Sample size, denoted by 'n' in our exercise, significantly affects the calculation of confidence intervals. Larger sample sizes tend to yield more precise estimates of the population parameter, resulting in narrower confidence intervals. This is because they provide more data points, thus reflecting the population more accurately and reducing the margin of error.

In plain terms, if you want to know what kind of pizza toppings a city prefers, asking 500 people will provide a clearer picture than asking just 5. Therefore, when we discuss the confidence interval becoming narrower with a larger sample size, it's akin to sharpening the focus on our pizza preference picture; details become clearer and more reliable.

Proportion

Proportion in our exercise refers to the part of the surveyed sample that represents a particular characteristic—like the proportion of students attending their first choice of college. A proportion is a type of ratio, which can be expressed as a fraction, percentage, or decimal, and is fundamental in calculating the confidence interval.

Imagery can sometimes simplify the understanding of a proportion. Think of it as slicing a pie in accordance with who wants a piece. In our case, 72.1% is akin to cutting the pie so that seven out of ten friends ideally get the exact subpart they want. When you work with proportions in statistics, you are allocating parts of a whole based on certain criteria or responses, just as you would divide a pie to satisfy varying appetites.