Question

The fraternities and sororities at St. Algebra College have been plagued by declining membership over the past several years and want to know whether 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 be expressed as numbers?)

Short answer

271 to 476 freshmen are estimated to be extremely interested in social clubs.

Step-by-step solution

- Calculate the Sample Proportion

Calculate the proportion of freshmen in the sample who are extremely interested in social clubs. This can be done using the formula: \[ \text{Sample Proportion} = \frac{\text{Number of Interested Respondents}}{\text{Total Number of Respondents}} \]Here, \(\text{Number of Interested Respondents} = 35\) and \(\text{Total Number of Respondents} = 150\). Substitute these values into the formula to get: \[ \text{Sample Proportion} = \frac{35}{150} = 0.2333 \]

- Find the Standard Error

Calculate the standard error of the sample proportion using the formula: \[ \text{Standard Error} = \frac{\text{Sample Proportion} \times (1 - \text{Sample Proportion})}{\text{Total Number of Respondents}} \] Substitute the sample proportion from Step 1. \[ \text{Standard Error} = \frac{0.2333 \times (1 - 0.2333)}{150} \]This gives us: \[ \text{Standard Error} \thickapprox 0.0326 \]

- Determine the Z-value

For a 95% confidence level, the Z-value is approximately 1.96. This value is taken from the standard normal distribution table.

- Calculate the Confidence Interval

Calculate the confidence interval for the sample proportion using the formula: \[ \text{Confidence Interval} = \text{Sample Proportion} \thickspace \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\text{-}}}}}}}}}}}}}}}}}}}}}}}} \text{Z-value} \times \text{Standard Error} \]Substitute the values: \[ \text{Confidence Interval} = 0.2333 \thickspace \boldsymbol{-} 1.96 \times 0.0326 \boldsymbol{\boldsymbol{-}} 0.2333 \thickspace \boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{-}}}} 1.96 \times 0.0326 \]Calculate to get: \[ \text{Confidence Interval} = 0.2333 \thickspace \boldsymbol{-} 0.064 \text{-} 0.2333 + 0.064 \] This gives the interval: \[ 0.1693 \thickspace \boldsymbol{\boldsymbol{\text{-}}} 0.2973 \]

- Calculate the Estimates for Total Freshmen

Finally, convert the proportions to the actual number of students by multiplying by the total number of freshmen: \[ \text{Estimate} = \text{Proportion} \times \text{Total Number of Freshmen} \]Substitute the extreme ends of the interval: \[ \text{Low Estimate} = 0.1693 \times 1600 = 270.88 \]\[ \text{High Estimate} = 0.2973 \times 1600 = 475.68 \]So, the estimates are approximately 271 and 476 freshmen.

Key concepts

Sample Proportion

In statistics, the **sample proportion** is a key concept when estimating population parameters based on a sample. It's simply the ratio of members in a sample displaying a particular characteristic to the total number of members in the sample. For this exercise, to determine the **sample proportion** of freshmen extremely interested in social clubs, we use the formula: \( \text{Sample Proportion} = \frac{\text{Number of Interested Respondents}}{\text{Total Number of Respondents}} \)
Here, 35 out of 150 respondents are extremely interested. By substituting these values into the formula, we get: \( \text{Sample Proportion} = \frac{35}{150} = 0.2333 \). This tells us that approximately 23.33% of the sampled freshmen express extreme interest. Understanding this proportion is essential as it forms the foundation for further calculations.

Standard Error Calculation

The **standard error** plays a crucial role in gauging the accuracy of the sample proportion as an estimate of the population proportion. It quantifies the variability of the sample proportion and is calculated using: \[ \text{Standard Error} = \sqrt{\frac{\text{Sample Proportion} \times (1 - \text{Sample Proportion})}{\text{Total Number of Respondents}}} \]
Employing our sample proportion of 0.2333, the calculation becomes: \[ \text{Standard Error} = \sqrt{\frac{0.2333 \times (1 - 0.2333)}{150}} \approx 0.0326 \]
This value reflects the expected variability in our proportion estimate if we could repeat the survey multiple times under the same conditions. A lower standard error indicates a more precise estimate.

Z-value

The **Z-value** corresponds to the number of standard errors a particular value is away from the mean in a standard normal distribution, with its use depending on the desired confidence level. For a **95% confidence level**, the Z-value often used is **1.96**. This value is derived from statistical tables of the standard normal distribution.
The importance of the Z-value lies in its role in constructing the confidence interval, thus giving a range in which we expect the true population proportion to fall with a certain level of confidence. Using this Z-value, we can account for the variability in our estimate and derive a more reliable prediction.

Confidence Interval

The **confidence interval** provides a range of values within which the true population parameter is likely to fall, given a specified level of confidence. We calculate the confidence interval for the sample proportion using: \[ \text{Confidence Interval} = \text{Sample Proportion} \pm \text{Z-value} \times \text{Standard Error} \]
For our sample proportion of 0.2333 and a standard error of 0.0326, the confidence interval, considering a Z-value of 1.96, is: \[ \text{Confidence Interval} = 0.2333 \pm 1.96 \times 0.0326 \]
This equates to approximately: \[ 0.1693 \text{ to } 0.2973 \].
To convert this interval into an estimate for the total number of freshmen, we multiply by the total number of freshmen (1600): \[ \text{Low Estimate} = 0.1693 \times 1600 = 270.88 \approx 271 \] \[ \text{High Estimate} = 0.2973 \times 1600 = 475.68 \approx 476 \]
Hence, we estimate that between **271 and 476 freshmen** might be extremely interested in social clubs. This range accounts for sampling variability and provides a realistic expectation of the true population proportion.