91Ó°ÊÓ

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)?

Step by step solution

01

Calculate the number of successes

A success in this context would denote a student who is attending their first choice of college. The proportion of successes is given as \(72.1\% = 0.721\). This means if there are 500 students, the number of successes would be \(0.721 * 500 = 361\).

02

Calculate the standard error

The standard error can be calculated using the formula \(\sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the proportion of successes and \(n\) is the total number of trials (or students in this context). So, our standard error would be \(\sqrt{\frac{0.721*(1-0.721)}{500}} = 0.0203\).

03

Calculate the 99% confidence interval

The 99% confidence interval can be calculated using the formula \(p \pm Z*SE\), where \(Z\) is the Z-value corresponding to the required confidence level (in this case, 99%). For a 99% confidence level, the Z-value is approximately 2.576. Therefore the confidence interval is \(0.721 \pm 2.576*0.0203 = (0.676, 0.766)\). So we could say we are 99% confident that the true proportion of college students who are attending their first choice college falls within the interval (67.6%, 76.6%).

04

Calculate failures

Those who are not attending their first choice college are considered failures. This is \(1 - 0.721 = 0.279\). So, with 500 students, the number of failures would be \(0.279 * 500 = 139.5\).

05

Calculate standard error for failures

Applying the same formula as before we get \(SE = \sqrt{\frac{0.279*(1-0.279)}{500}} = 0.0203\).

06

Calculate the 95% confidence interval for failures

For a 95% confidence level, the Z-value is approximately 1.96. So the confidence interval becomes \(0.279 \pm 1.96*0.0203 = (0.240, 0.318)\). Thus, we are 95% confident that the true proportion of students who are not attending their first choice falls within the interval (24.0%, 31.8%).

07

Analyzing the effect of increased sample size

As the sample size increases, the standard error of the proportion decreases. This means the width of the confidence interval would be narrower with a larger sample size.

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Key Concepts

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

Proportion Calculation

Calculating proportions is a foundational step in statistical analysis. In the context of the exercise, a proportion refers to the fraction of students attending their first-choice college.
This is expressed as a percentage of the total sample size. For example, if 72.1% of students in the survey are attending their first choice, this means 0.721 is the proportion of success.
To find the number of such students, multiply the total number of students surveyed by this proportion. Thus, with 500 students surveyed, we calculate this as:

• \[0.721 \times 500 = 361\]

This numerical value helps in further calculations, such as determining the confidence interval for this proportion.

Standard Error

The standard error (SE) plays a critical role in estimating how much an observed sample proportion will differ from the true population proportion. It provides a sense of how spread out the sample proportions are.
For confusion reduction, the formula to calculate the standard error for a proportion is:

• \[SE = \sqrt{\frac{p(1-p)}{n}}\]

Here, \(p\) is the sample proportion, and \(n\) is the sample size. For our example, with \(p = 0.721\) and \(n = 500\):

• \[SE = \sqrt{\frac{0.721 \times (1-0.721)}{500}} \approx 0.0203\]

This value indicates how much the sample proportion is expected to vary from one sample to another, guiding us in constructing confidence intervals.

Sample Size Impact

The impact of sample size on statistical measures, particularly confidence intervals, is profound. A larger sample size tends to provide more accurate estimates of the population parameters.
When calculating confidence intervals, the sample size affects the width of the interval. Specifically, a larger sample size results in a smaller standard error.

• Smaller SE leads to narrower confidence intervals.
• This makes our estimate more precise and reliable.

Hence, in our exercise, if the actual sample size was larger than 500, the confidence interval would be narrower. This would indicate a more precise estimation of the proportion of students attending their first-choice schools. Larger sample sizes thus enhance the accuracy of the predictions we make from our data.