Question

The fraternities and sororities at St. Algebra College have been plagued by declining membership over the past several years and want to know if the incoming freshman class will be a fertile recruiting ground. Not having enough money to survey all 1600 freshmen, they commission you to survey the interests of a random sample. You find that 35 of your 150 respondents are "extremely" interested in social clubs. At the \(95 \%\) level, what is your estimate of the number of freshmen who would be extremely interested? (HINT: The high and low values of your final confidence interval are proportions. How can proportions also be expressed as numbers?)

Short answer

Approximately 264 to 483 freshmen are estimated to be extremely interested in social clubs at the 95% confidence level.

Step-by-step solution

- Understanding the Problem

Determine the problem and what needs to be found. A random sample of 150 students found that 35 are 'extremely' interested in social clubs. The task is to estimate, at the 95% confidence level, how many of the 1600 freshmen are likely to be extremely interested in social clubs.

- Determine the Sample Proportion

Calculate the sample proportion (p̂). The sample proportion is the number of students extremely interested divided by the total number of respondents. p̂ = \( \frac{35}{150} = 0.2333 \)

- Calculate the Standard Error

Calculate the standard error (SE) of the sample proportion using the formula: SE = \( \sqrt{ \frac{p̂(1 - p̂)}{n} } \) where n is the sample size. SE = \( \sqrt{ \frac{0.2333 (1 - 0.2333)}{150} } \) SE ≈ 0.0350

- Find the Z-Score for 95% Confidence Interval

Find the Z-score for the 95% confidence level. From the Z-table, the Z-score corresponding to a 95% confidence level is approximately 1.96.

- Calculate the Confidence Interval

Calculate the margin of error (MoE) using the formula: MoE = Z × SE MoE = 1.96 × 0.0350 ≈ 0.0686 Next, find the confidence interval for p̂: Lower limit = p̂ - MoE ≈ 0.2333 - 0.0686 ≈ 0.1647 Upper limit = p̂ + MoE ≈ 0.2333 + 0.0686 ≈ 0.3019 The 95% confidence interval for the proportion is (0.1647, 0.3019).

- Convert Proportions to Numbers

Convert the proportions to the estimated numbers of freshman. Lower estimate = 0.1647 × 1600 ≈ 263.52 Upper estimate = 0.3019 × 1600 ≈ 483.04 Therefore, the 95% confidence interval for the number of freshmen who are extremely interested in social clubs is approximately (264, 483).

Key concepts

Sample Proportion

The sample proportion, often denoted as \( \hat{p} \) (p-hat), is a key concept in statistics. It represents the ratio of the observed characteristic in your sample to the total sample size. In the context of the problem, the sample proportion was calculated by dividing the number of students who are 'extremely' interested in social clubs (35) by the total number of respondents (150). This gives you a sample proportion \( \hat{p} = \frac{35}{150} \approx 0.2333 \). This is essentially an estimate of the true proportion in the entire population based on your sample.

Standard Error

The standard error (SE) measures the variability or standard deviation of a sampling distribution. It tells you how much the sample proportion is expected to fluctuate from the true population proportion. The formula for calculating the standard error of a sample proportion is \( SE = \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{ n } } \). In the exercise, we use the sample proportion \( \hat{p} = 0.2333 \) and the sample size \( n = 150 \), giving us \( SE \approx 0.0350 \.\) A smaller standard error means that there’s less variability in your estimates.

Margin of Error

The margin of error (MoE) gives you a range that is likely to contain the true population proportion. It is calculated by multiplying the standard error by the Z-score associated with your desired confidence level. For a 95% confidence level, the Z-score is 1.96. Thus, \( MoE = Z \times SE = 1.96 \times 0.0350 \approx 0.0686 \). This means that the sample proportion can vary by this margin of error on either side. It provides a measure of the precision of your sample estimate.

Z-Score

A Z-score represents the number of standard deviations a data point (or proportion) is from the mean of a distribution. For confidence intervals, the Z-score corresponding to a 95% confidence level is approximately 1.96. This score is derived from the standard normal distribution, which is why it is often referred to as a standard score. In confidence interval estimation, we use the Z-score to determine the extent of deviation from the sample proportion, which provides the basis for our margin of error.

Random Sampling

Random sampling is a technique where every member of the population has an equal chance of being selected in the sample. This ensures that the sample is representative of the entire population, minimizing bias. In the provided exercise, the random sample consisted of 150 freshmen out of a total of 1600. Using random sampling increases the reliability of the results and guarantees that the findings can be generalized to the entire population. This is crucial for making valid inferences from your data.