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In this section, we assumed that the sample size was less than \(5 \%\) of the size of the population. When sampling without replacement from a finite population in which \(n>0.05 N,\) the standard deviation of the distribution of \(\hat{p}\) is given by $$\sigma_{p}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n-1} \cdot\left(\frac{N-n}{N}\right)}$$ where \(N\) is the size of the population. Suppose a survey is conducted at a college having an enrollment of 6,502 students. The student council wants to estimate the percentage of students in favor of establishing a student union. In a random sample of 500 students, it was determined that 410 were in favor of establishing a student union. (a) Obtain the sample proportion, \(\hat{p},\) of students surveyed who favor establishing a student union. (b) Calculate the standard deviation of the sampling distribution of \(\hat{p}\)

Step by step solution

01

Calculate Sample Proportion

First, find the sample proportion \(\hat{p}\). This is calculated as the number of students in favor of establishing a student union divided by the total number of students surveyed. \(\hat{p} = \frac{410}{500} = 0.82\)

02

Substitute Values into the Formula

Next, substitute the sample size \(n = 500\), the population size \(N = 6502\), and the sample proportion \(\hat{p} = 0.82\) into the standard deviation formula: \(\sigma_{\hat{p}} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n - 1} \cdot \left(\frac{N - n}{N}\right)}\)

03

Calculate Each Component

Calculate each part of the formula separately: \(\hat{p}(1-\hat{p}) = 0.82 \cdot (1 - 0.82) = 0.1476\), \(\frac{N - n}{N} = \frac{6502 - 500}{6502} = \frac{6002}{6502} \approx 0.92299\)

04

Combine and Compute Standard Deviation

Now combine these components into the formula and compute: \(\frac{0.1476}{499} \approx 0.000296\), \(\sigma_{\hat{p}} = \sqrt{0.000296 \cdot 0.92299} \approx \sqrt{0.000273} \approx 0.0165\)

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Key Concepts

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

Sample Size

Sample size, denoted as \(n\), is the number of individual observations or data points collected in a survey or an experiment. It plays a crucial role in determining the accuracy and reliability of the statistical conclusions.

• A larger sample size generally leads to more precise estimates of the population parameters.
• It reduces sampling variability, meaning the results are more consistent across different samples.
• Sample size should be selected keeping practical constraints like time, cost, and available population in mind.

In the exercise, the sample size was 500 out of a total population of 6,502 students. A well-chosen sample size ensures the standard deviation calculation is meaningful and accurately reflects population characteristics.

Standard Deviation

Standard deviation is a measure of the dispersion or variability within a set of values. It indicates how much individual data points deviate from the mean of the dataset. In a sampling distribution, it measures how much the sample proportion might vary from the true population proportion

• It is computed using the formula for the standard deviation of a sample proportion: \( \sigma_{\hat{p}} = \sqrt{\frac{\big( \hat{p}(1 - \hat{p}) \big )}{n - 1} \cdot \left ( \frac{N - n}{N} \right) } \)
• High standard deviation means data points are spread out over a wider range.
• Low standard deviation signifies data points are closer to the mean.

For our exercise, a standard deviation of approximately 0.0165 was calculated, which provides insight into the variability of the sample proportion.

Sample Proportion

The sample proportion, denoted as \(\hat{p}\), represents the ratio of individuals displaying a certain characteristic to the total number of individuals in the sample.

• It is calculated by dividing the number of favorable outcomes by the total number of observations.
• It provides an estimate of the population proportion if the sample is representative.
• Sample proportion helps in creating confidence intervals and conducting hypothesis tests.

In the given exercise, the sample proportion of students supporting the union was calculated as \( \frac{410}{500} = 0.82 \). This means 82% of the sampled students favored the establishment of a student union.

Finite Population Correction

Finite population correction (FPC) is a factor applied to statistical calculations when a significant fraction of the population is sampled. It adjusts the standard deviation to account for the reduced variability in larger samples from finite populations.

• This correction is only used when the sample size is more than 5% of the total population.
• FPC reduces the standard deviation because sampling without replacement leads to less variability.
• The correction factor is calculated as \(\sqrt{\frac{N - n}{N}} \), where \(N\) is the population size and \(n\) is the sample size.

In the exercise, the FPC was calculated as \( \sqrt{\frac{6002}{6502}} \approx 0.92299 \), reflecting how sampling from this finite population influences the variability of our sample proportion.