Question

The article "Thrillers" (Newsweek, April 22,1985 ) stated, "Surveys tell us that more than half of America's college graduates are avid readers of mystery novels." Let \(p\) denote the actual proportion of college graduates who are avid readers of mystery novels. Consider a sample proportion \(\hat{p}\) that is based on a random sample of 225 college graduates. If \(p=0.5,\) what are the mean value and standard deviation of the sampling distribution of \(\hat{p}\) ? Answer this question for \(p=0.6 .\) Is the sampling distribution of \(\hat{p}\) approximately normal in both cases? Explain.

Short answer

For p=0.5, the mean and standard deviation of the sampling distribution of \(\hat{p}\) are 0.5 and \(\sqrt{(0.5)(0.5)/225}\), respectively. For p=0.6, the mean and standard deviation are 0.6 and \(\sqrt{(0.6)(0.4)/225}\), respectively. In both cases, because \(np\) and \(n(1-p)\) values are greater than 5, the sampling distributions are approximately normal.

Step-by-step solution

Determine Mean Value

First, we determine the expected mean value of the sampling distribution of the proportion. We use the formula for the mean of the sampling distribution of proportions, which is simply the population proportion. So, for p=0.5, \(\mu = p\). For p=0.6, \(\mu = p\) as well.

Determine Standard Deviation

Next, we calculate the standard deviation of the sampling distribution of the proportion, denoted by \( \sigma \). Use the formula: \(\sigma = \sqrt{pq/n}\), where p is the population proportion, q is (1-p), and n is the sample size. Note that this formula assumes that the Central Limit Theorem applies (n is large enough). So, compute \( \sigma \) for both cases, p=0.5 and p=0.6.

Check Normality

Finally, we need to verify whether the sampling distribution of the proportion could be considered as approximately normal. According to the Central Limit Theorem, the sampling distribution can be approximated to a normal distribution if both \(np\) and \(n(1-p)\) are greater than or equal to 5. Therefore, calculate \(np\) and \(n(1-p)\) for both p=0.5 and p=0.6, and see if they are greater than or equal to 5. If they are, the distribution would be approximately normal.

Key concepts

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in the realm of statistics. It tells us that when an adequately large sample size is taken from a population, the distribution of the sample means will approximate a normal distribution, regardless of the shape of the population distribution. This approximation becomes better with larger sample sizes.

For a sample proportion, the CLT implies that the sampling distribution of the proportion \( \hat{p} \) will become normally distributed as the sample size increases, given that certain conditions are met. These conditions include having a sufficiently large sample size to satisfy \(np \geq 5\) and \(n(1-p) \geq 5\), where \(n\) is the sample size and \(p\) is the population proportion. In our case, with a sample size of 225 college graduates, we check these conditions to ensure that the distribution of \( \hat{p} \) is normal for the given proportions (0.5 and 0.6).

Understanding the CLT is crucial because it allows us to make inferences about a population from our sample data, and it is the reasoning behind many statistical tests and confidence intervals.

Population Proportion

Population proportion, denoted by \(p\), is a measure that represents the fraction of the population that exhibits a certain characteristic. For example, if \(p=0.5\), it indicates that half of the population is considered to have the attribute in question—in our exercise, being avid readers of mystery novels.

In practical terms, \(p\) is not always known and is often estimated using sample data. The closer the sample proportion \(\hat{p}\) is to \(p\), the better our estimate of the population attribute. When calculating the mean of the sampling distribution of sample proportions, the mean is equivalent to \(p\), the true population proportion. Thus, if the actual proportion of college graduates who are avid mystery novel readers is 0.5, our mean \(\mu\) for the sampling distribution will also be 0.5. This remains true for any value of \(p\), such as 0.6 in the exercise provided.

Standard Deviation of a Proportion

The standard deviation of a proportion, symbolized as \(\sigma\), refers to the measure of variability in the sample proportions from the true population proportion. It quantifies the dispersion of the sampling distribution of proportions around the population proportion \(p\). The formula to calculate this standard deviation is given by \(\sigma = \sqrt{\frac{pq}{n}}\), where \(q\) is the complement of the population proportion \(1-p\), and \(n\) is the sample size.

For instance, to find the standard deviation of the proportion for the assumption that \(p=0.5\), we would plug in these values into our formula—\(q = 1 - 0.5 = 0.5\) and \(n = 225\). A similar calculation would be done for \(p=0.6\). Knowing the precise value of the standard deviation allows us to understand how much the sample proportion can be expected to vary, which is crucial for tasks such as constructing confidence intervals or conducting hypothesis tests.