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You'd like to estimate the proportion of the 14,201 undergraduate students at Syracuse University who are full-time students. You poll a random sample of 350 students, of whom 330 are full-time. Unknown to you, the proportion of all undergraduate students who are full-time students is \(0.951 .\) Let \(X\) denote a random variable for which \(x=1\) denotes full-time student and for which \(x=0\) denotes part-time student. (For recent enrollment numbers, go to www.syr.edu/about/ facts.html.) a. Describe the population distribution. Sketch a graph representing the population distribution. b. Describe the data distribution. Sketch a graph representing the data distribution. c. Find the mean and standard deviation of the sampling distribution of the sample proportion for a sample of size \(350 .\) Explain what this sampling distribution represents. Sketch a graph representing this sampling distribution. d. Use the Sampling Distribution app accessible from the book's website to check your answers from parts a through \(\mathrm{c}\). Set the population proportion equal to \(p=0.951\) and \(n=350 .\) Compare the population, data, and sampling distribution graph from the app with your graphs from parts a through \(\mathrm{c}\).

Step by step solution

01

Describe the Population Distribution

The population distribution reflects the status of all 14,201 undergraduate students at Syracuse University concerning their enrollment as full-time or part-time. The population proportion of full-time students is given as \( p = 0.951 \). This means 95.1% of undergraduate students are full-time. The distribution is categorical with possible values of 0 or 1, but since 95.1% are full-time, the '1' value (full-time) would be much more frequent. A bar graph would show a much taller bar for full-time students compared to part-time students.

02

Describe the Data Distribution

The data distribution, based on a sample of 350 students, has 330 full-time students and 20 part-time students. Thus, the sample proportion of full-time students is \( \hat{p} = \frac{330}{350} = 0.943 \). Like the population distribution, this is also categorical and shows a distribution with a strong preference for full-time students, but slightly less so than in the population distribution.

03

Mean of the Sampling Distribution

The mean of the sampling distribution of the sample proportion \( \hat{p} \) equals the population proportion \( p \). Thus, \( \mu_{\hat{p}} = p = 0.951 \). This represents the expected proportion of full-time students in all random samples of size 350, repeated infinitely.

04

Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution, known as the standard error, is calculated using the formula \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \). Substituting the values, \( \sigma_{\hat{p}} = \sqrt{\frac{0.951 \times 0.049}{350}} \approx 0.0114 \). This measures the variability of the sample proportion from the population proportion when using a sample size of 350.

05

Explanation and Graph of Sampling Distribution

The sampling distribution of the sample proportion is approximately normal due to the large sample size (Central Limit Theorem). Its mean is \( 0.951 \) and standard deviation is \( 0.0114 \). The graph would be a normal distribution curve centered around \( p = 0.951 \) with a relatively narrow spread due to the small standard deviation.

06

Use the Sampling Distribution App

Access the Sampling Distribution app, set the population proportion to \( p = 0.951 \), and the sample size \( n = 350 \). Observe the app-generated graphs for the population distribution, data distribution, and sample proportion distribution. Compare these with your sketches from Steps 1 to 5 to confirm their accuracy and consistency.

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Key Concepts

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

Understanding Population Distribution

In statistics, a population distribution represents the entire set of data or outcomes for a specific population. In our exercise, it's about the 14,201 undergraduate students at Syracuse University. This distribution shows us how these students are classified as either full-time or part-time. Here, the population proportion of full-time students is substantial, specifically given as 0.951. This means that 95.1% of the students are full-time.
This type of distribution is categorical because it's about counting distinct categories rather than numerical values. For this case, the categories are "full-time" or "part-time", represented by numbers 1 and 0 respectively. If we illustrated this on a bar graph, you'd see significantly more students under the "full-time" category, creating a much taller bar compared to the "part-time" category.
Understanding this distribution helps us grasp the overall trend within the university’s student body and sets the stage for further statistical analysis in segments like sampling distributions.

Exploring Standard Error

The standard error is a significant concept in statistics, specifically when dealing with samples. It refers to the measure of the variability of a sample statistic from the true population statistic. In simpler terms, it tells us how much our sample mean (or proportion) is expected to fluctuate due to the randomness of sampling.
For this exercise, we calculate the standard error of the sample proportion to understand how the proportion of full-time students in a sample may differ from the proportion in the entire population. The formula used is \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \] where \( p \) is the population proportion, and \( n \) is the sample size. Here, substituting in our values, \( p = 0.951 \) and \( n = 350 \), we find the standard error to approximately be 0.0114.
This minor standard error reflects that with a large sample, our sample proportion closely represents the true proportion. It assures us that new surveys or samples would give similar insights into the population's enrollment ratio, enhancing the reliability of the sampled data.

The Central Limit Theorem Unveiled

The Central Limit Theorem (CLT) is a fundamental concept in statistics that helps justify the normal approximation of sampling distributions. It states that when you take sufficiently large samples from a population, the distribution of sample proportions will be approximately normally distributed, regardless of the shape of the population distribution.
In our context, the Central Limit Theorem supports the assumption that the sampling distribution of the sample proportion is bell-shaped and centered around the population proportion \( p = 0.951 \). This happens because our sample size of 350 is large enough to ensure the normality of the sampling distribution.
The CLT is crucial because it allows us to make inferences about a population from a sample. It means we can presume that our calculated proportions from such samples are reliable predictors of the overall population characteristics. Thus, with a mean of 0.951 and a standard deviation calculated as the standard error of 0.0114, our sampling distribution appears as a normal distribution with a narrow spread, offering confidence in the statistical insights derived from the gathered sample.