Question

Consider interviewing a random sample of \(n=50\) adults. Let \(\hat{P}\) denote the proportion of the 50 sampled adults who drink coffee. If the population proportion of coffee drinkers is \(0.80,\) what is the appropriate approximate model for the distribution of \(\hat{P}\) over many such samples of size \(50 ?\) That is, what type of distribution is this, what is the mean, and what is the standard deviation?

Short answer

The sampling distribution of \(\hat{P}\) is approximately normal with a mean of 0.80 and a standard deviation of 0.0566.

Step-by-step solution

Determine the Distribution Type

Since we're sampling proportions and the sample size is large enough, we assume the sampling distribution of the sample proportion \(\hat{P}\) is approximately normal due to the Central Limit Theorem (CLT). To use the normal approximation, the sample size must satisfy the conditions \(np \geq 10\) and \(n(1-p) \geq 10\).

Verify CLT Conditions for Normal Approximation

Check if the sample size and the population proportion satisfy the conditions for normal approximation: \(n \cdot p = 50 \cdot 0.80 = 40\) and \(n \cdot (1-p) = 50 \cdot 0.20 = 10\), both of which are greater than or equal to 10. Therefore, the normal approximation is appropriate.

Calculate the Mean of the Sampling Distribution

The mean of the sampling distribution of \(\hat{P}\) is equal to the population proportion, \(\mu_{\hat{P}} = p = 0.80\).

Calculate the Standard Deviation of the Sampling Distribution

The standard deviation of \(\hat{P}\) is calculated by the formula \(\sigma_{\hat{P}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.80 \cdot 0.20}{50}} = \sqrt{\frac{0.16}{50}} = \sqrt{0.0032} = 0.0566\). To two decimal places, the standard deviation is 0.0566.

Key concepts

Central Limit Theorem

The Central Limit Theorem (CLT) is a foundational concept in statistics that demonstrates how the distribution of sample means will approach a normal distribution as the sample size grows, regardless of the original distribution of the population. When we collect samples and calculate averages or proportions, the CLT guarantees that these statistics will form a bell-shaped curve if the sample size is large enough.

For example, when you repeatedly take samples of 50 adults to find out the proportion who drink coffee, these proportions will tend to form a normal distribution around the true population mean, assuming you have enough samples. The magic number often cited for 'large enough' is 30, but in practice, it can vary. Nonetheless, a sample size of 50, as given in our exercise, is sufficient to invoke the CLT for a normal approximation.

Normal Approximation

Normal approximation refers to the process of using a normal distribution to model a different, non-normal distribution. This method is often used when dealing with sampling distributions of proportions or means because, thanks to the Central Limit Theorem, we know that these distributions will approximate normality. The key criteria for this approximation to be valid are that the conditions set by the CLT are met – namely, that both the number of successes (np) and failures (n(1-p)) in the samples are greater than or equal to 10.

In our case, since the sample size n is 50 and the population proportion p is 0.80, both the number of expected successes (np = 40) and failures (n(1-p) = 10) exceed this threshold. This justifies the use of a normal distribution to approximate the behavior of the sample proportion \(\hat{P}\) over several samples.

Population Proportion

The population proportion is a measure of the frequency with which a particular trait or characteristic occurs in the population. It is denoted as p in statistical formulas. In our exercise, the population proportion (p) is the true proportion of adults who drink coffee, which is given as 0.80 or 80%. This figure is used to calculate the mean of the sampling distribution and assess whether the sample size is adequate to use a normal approximation for the distribution of the sample proportion, \(\hat{P}\).

Standard Deviation

Standard deviation is a statistic that quantifies the amount of dispersion in a set of values. In the context of sampling distributions, the standard deviation of the sample proportion (\(\sigma_{\hat{P}}\)) tells us how much we would expect the proportion of coffee drinkers in a sample to vary from the population proportion. The formula for calculating this for a proportion is \[\sigma_{\hat{P}} = \sqrt{\frac{p(1-p)}{n}}\]

For our given problem, the standard deviation helps in determining how spread out the sample proportions are around the population proportion of 0.80. The smaller the standard deviation, the closer the sample proportions are likely to cluster around the population proportion.

Sample Proportion

The sample proportion (\(\hat{P}\)) is the estimate of the population proportion based on a particular sample. It is found by dividing the number of successes (in this case, coffee drinkers) by the total number of individuals in the sample. When we talk about the sampling distribution of \(\hat{P}\), we're referring to the probability distribution that would be formed if we took an infinite number of samples of the same size from the population and calculated \(\hat{P}\) each time.

This concept is central to inferential statistics as it allows us to make estimations or predictions about the population proportion based on sample data. By using the sample proportion along with the other concepts, such as standard deviation and using the Central Limit Theorem for normal approximation, we can calculate probabilities and make confidence statements about the population.