Question

A West Coast university has found that about \(90 \%\) of its accepted applicants for enrollment in the freshman class will actually enroll. In 2012,1360 applicants were accepted to the university. Within what limits would you expect to find the size of the freshman class at this university in the fall of \(2012 ?\)

Short answer

Answer: We expect the size of the freshman class to be within the limits of 1202 and 1246 students.

Step-by-step solution

Calculate the mean and standard deviation

First, we need to calculate the mean and standard deviation for the binomial distribution using the given information: Number of trials (accepted applicants), \(n = 1360\) Probability of success (percentage enrolled), \(p = 0.90\) Mean, \(\mu = n * p = 1360 * 0.90 = 1224\) Standard deviation, \(\sigma = \sqrt{n * p * (1-p)} = \sqrt{1360 * 0.90 * (1-0.90)} = \sqrt{1360 * 0.90 * 0.10} = \sqrt{122.4} \approx 11.06\) The mean and standard deviation for the binomial distribution are 1224 and 11.06, respectively.

Determine the limits using the empirical rule

Next, we will use the empirical rule to determine the limits within which we expect the size of the freshman class to be. To do this, we need to find the range within two standard deviations of the mean (95% confidence interval). Lower limit: \(\mu - 2\sigma = 1224 - 2(11.06) = 1224 - 22.12 \approx 1201.88\) Upper limit: \(\mu + 2\sigma = 1224 + 2(11.06) = 1224 + 22.12 \approx 1246.12\)

Round the limits to the nearest whole number

Since we cannot have a fraction of a student enrolled, we need to round the limits to the nearest whole number. Lower limit: \(1201.88\) rounded to the nearest whole number is \(1202\) Upper limit: \(1246.12\) rounded to the nearest whole number is \(1246\) We would expect the size of the freshman class at this university in the fall of 2012 to be within the limits of 1202 and 1246 students.

Key concepts

Empirical Rule

The Empirical Rule is a helpful guideline in statistics, especially when dealing with normal distributions. It tells us about the spread of data within a certain number of standard deviations from the mean. This rule is particularly useful to quickly understand the variability without calculating each point individually.
For any bell-shaped, or normally distributed data:

• About 68% of data falls within one standard deviation (\(\sigma\)) of the mean.
• About 95% of data falls within two standard deviations.
• Approximately 99.7% is within three standard deviations.

In the context of our exercise, we use the empirical rule to estimate where most of the university's freshman class enrollment numbers will fall. By calculating two standard deviations from the mean, we find that 95% of the freshman class sizes will likely be between 1202 and 1246 students. This offers a high-level overview of what to generally expect without getting lost in complex calculations.

Confidence Interval

A Confidence Interval provides a range of values, derived from sample data, within which we can be "confident" that the true population parameter lies. Unlike a precise point estimate, it expresses uncertainty and variability. This is crucial whenever we deal with samples instead of whole populations.
In our problem, we're estimating the size of the freshman class by calculating the confidence interval around the mean number of enrollments. We used the 95% confidence interval, covering roughly two standard deviations from the mean.
To establish this interval:

• Calculate the mean (\(\mu\)).
• Determine the standard deviation (\(\sigma\)).
• Use the empirical rule to find the interval: \(\mu \pm 2\sigma\).

Thus, we estimate the freshman class size to be within the range of 1202 to 1246 students, encapsulating the likely variation in a meaningful way.

Standard Deviation

Standard Deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of values. It shows how much individual data points differ from the mean. A small standard deviation means data points are close to the mean, while a large one indicates a wider spread.
In the problem at hand, the standard deviation (\(\sigma\)) tells us how much the number of enrolled freshmen might vary from the average calculated mean of 1224.
To calculate it for the binomial distribution, we used the formula:\[ \sigma = \sqrt{n \times p \times (1-p)} \]Here:

• \(n\) = number of trials (1360 accepted applicants).
• \(p\) = probability of enrollment (0.90).

This gives us a standard deviation of about 11.06. Understanding this helps identify that most enrollment numbers will fall within two times this amount (i.e., in the range of 1202 to 1246) around the mean, according to the empirical rule.