Question

The article "Students Increasingly Turn to Credit Cards" (San Luis Obispo Tribune, July 21,2006 ) reported that \(37 \%\) of college freshmen carry a credit card balance from month to month. Suppose that the reported percentage was based on a random sample of 1,000 college freshmen. Suppose you are interested in learning about the value of \(p,\) the proportion of all college freshmen who carry a credit card balance from month to month. The following table is similar to the table that appears in Examples 8.4 and \(8.5,\) and is meant to summarize what you know about the sampling distribution of \(\hat{p}\) in the situation just described. The "What You Know" information has been provided. Complete the table by filling in the "How You Know It" column.

Short answer

According to the Central Limit Theorem, for large sample sizes, the sampling distribution of the proportion \(\hat{p}\) is approximately normal, with mean equal to the true population proportion \(p\) and standard deviation calculated as \(\sqrt{\frac{p(1-p)}{n}}\). In this situation, the sample proportion \(\hat{p}\) is 0.37 and the sample size \(n\) is 1,000. Therefore, the estimated mean of the sampling distribution is 0.37, and the standard deviation can be calculated using the given formula. This information helps understand the characteristics of the sampling distribution of \(\hat{p}\) in this scenario.

Step-by-step solution

Determine sample size and sample proportion

The sample size \(n\) is given as 1,000 college freshmen and the sample proportion \(\hat{p}\) (proportion of freshmen who carry a credit card balance) is given as 0.37 or 37%.

Understand the sampling distribution of \(\hat{p}\)

The sampling distribution of \(\hat{p}\) is a theoretical probability distribution that would result if all possible samples of size \(n\) were drawn from the population and \(\hat{p}\) was calculated for each sample. By the Central Limit Theorem, if \(n\) is large enough, then the sampling distribution of \(\hat{p}\) is approximately normal.

Table Completion - Mean of \(\hat{p}\)

The mean of the sampling distribution of \(\hat{p}\) is equal to \(p\). This is known because it's a property of sampling distributions that the mean of the sampling distribution (expected value of \(\hat{p}\)) is equal to the true population proportion \(p\). Here, we do not know \(p\), but based on the sample, we can estimate \(p\) to be 0.37.

Table Completion - Standard deviation of \(\hat{p}\)

The standard deviation of the sampling distribution of \(\hat{p}\) can be calculated using the formula \(\sqrt{\frac{p(1-p)}{n}}\). This is a property of sampling distributions and it provides a measure of the dispersion or variability in the sampling distribution. Here, we can estimate it using the given sample proportion and size, substituting \(p = 0.37\) and \(n = 1000\) into the equation to get the estimated standard deviation.

Table Completion - Shape of the distribution

The shape of distribution is approximately normal. This is known because of the Central Limit Theorem which states that for large sample sizes, the sampling distribution is approximately normal.

Key concepts

Central Limit Theorem

Understanding the Central Limit Theorem (CLT) is crucial when dealing with statistics, especially sampling distributions. The CLT tells us that if we were to take a very large number of samples from a population, the sampling distribution of the sample means, or proportions, will approximate a normal distribution. This holds true regardless of the population's distribution shape, provided the sample size is sufficiently large, typically considered to be more than 30 samples.

Now, why does this matter? The Central Limit Theorem allows statisticians to make inferences about population parameters, such as means and proportions, using the sample statistics. It also underpins many statistical procedures, including hypothesis testing and confidence intervals. In our exercise, because the sample size is 1,000, which is well above 30, the sampling distribution of the sample proportion (the percentage of college freshmen carrying a credit card balance) will likely resemble a normal distribution, thanks to the CLT.

Sample Proportion

The sample proportion, typically denoted as \(\hat{p}\), represents the proportion of subjects in a sample that have a particular attribute. For example, in our exercise, the sample proportion is the percentage of sampled college freshmen who carry a credit card balance from month to month. Calculating the sample proportion is straightforward: it is the number of subjects with the attribute divided by the total sample size, which in this case is 37% from a sample of 1,000 students.

But the sample proportion is not just a number; it also serves as an estimator for the true population proportion, symbolized as \(p\). However, one must remember that \(\hat{p}\) can vary from sample to sample, and it's this variability that the sampling distribution seeks to describe and quantify.

Standard Deviation of Sampling Distribution

The standard deviation of the sampling distribution, often referred to as the standard error, measures the variation or spread of sample proportions \(\hat{p}\) around the population proportion \(p\). This standard deviation tells us how much we can expect the sample proportion to differ from the population proportion. It's an essential part of estimating margins of error and confidence intervals for percentages.

To calculate the standard deviation of the sampling distribution of \(\hat{p}\), we use the formula \(\sqrt{{\frac{{p(1-p)}}{{n}}}}\), where \(p\) is the population proportion and \(n\) is the sample size. In our exercise, since the population proportion \(p\) is unknown, we use the sample proportion as an estimate, leading us to \(\sqrt{{\frac{{0.37(1-0.37)}}{{1000}}}}\) to find the standard deviation of the sampling distribution.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a symmetrical bell-shaped curve that describes the spread of a dataset where the majority of the data points are concentrated around the mean. It's defined by two parameters: the mean and the standard deviation. The mean determines where the center of the curve is located, while the standard deviation determines the curve's spread.

In the context of sampling distributions, if the sample size is large enough, the central limit theorem ensures that the distribution of the sample proportion will be approximately normal. This is significant because it allows us to use standard statistical tables and tools that rely on normality assumptions. For example, we can use the z-table to find probabilities, standard scores, and critical values that are all important for tests of significance and constructing confidence intervals.