Question

The article "Facebook Etiquette at Work" (USA Today, March 24,2010 ) reported that \(56 \%\) of people participating in a survey of social network users said it was not \(\mathrm{OK}\) for someone to "friend" his or her boss. Let \(p\) denote the proportion of all social network users who feel this way and suppose that \(p=0.56\). a. Would \(\hat{p}\) based on a random sample of 50 social network users have a sampling distribution that is approximately normal? b. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) if the sample size is \(100 ?\)

Short answer

a. Yes, the sampling distribution of \(\hat{p}\) is approximately normal as both \(np\) and \(n(1-p)\) are greater than 10. b. The mean and standard deviation of the sampling distribution of \(\hat{p}\) if the sample size is 100 are 0.56 and 0.0498 respectively.

Step-by-step solution

Verify the normality of sampling distribution

Given that \(p=0.56\), n=50, and \(np=0.56*50= 28\) and \(n(1-p) = 50*0.44= 22 \). As both \(np\) and \(n(1-p)\) are greater than 10, we can conclude that the sampling distribution of \(\hat{p}\) is approximately normal.

Determine the mean of the sampling distribution

Using the formula \(\mu= p\), we can say that the mean is 0.56.

Determine the standard deviation of the sampling distribution

If the sample size is 100, the standard deviation can be calculated using the formula \(\sigma=\sqrt{\frac{p(1-p)}{n}}\). Therefore, \(\sigma=\sqrt{\frac{0.56*0.44}{100}}=0.0498\).

Key concepts

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that provides a deep insight into why many statistical methods work. It states that if you have a population with any distribution, the distribution of the average of a large number of independent random variables drawn from that population will be approximately normal (Gaussian). This is true no matter what the shape of the original distribution is, as long as the sample size is sufficiently large.

How large is 'sufficient'? A common rule of thumb is that the sample size should be at least 30. However, for more skewed distributions or ones with heavier tails, a larger sample size may be required to achieve normality. In our exercise involving the proportion of social network users who feel it's inappropriate to 'friend' a boss, the sample size in question is 50. According to CLT, since 50 is greater than 30, we would expect the sampling distribution of the proportion, denoted as \( \hat{p} \), to be approximately normal, given that the other normality conditions are also met.

Understanding the CLT helps students to grasp why we can perform hypothesis tests and create confidence intervals using normal distribution methods, even when dealing with proportions or other non-normal data. In practice, this theorem allows for the simplification of complex problems and the application of statistical techniques that are based on normal distribution.

Proportion of Social Network Users

When a study mentions the proportion of social network users who have a certain opinion or behavior, it refers to a parameter of the population. In our case, 'p' represents the true proportion of all social network users who believe it's not okay to 'friend' a boss. The value of \( p = 0.56 \) is an estimate from the survey results.

In statistics, we often need to estimate an unknown population parameter based on a sample. The proportion from a sample is denoted as \( \hat{p} \). When we collect sample data, we can calculate \( \hat{p} \) to make inferences about the true proportion 'p'. For instance, if we take a random sample of 100 social network users, we can determine \( \hat{p} \) for that sample. This proportion, however, might differ from the population proportion because of sampling variability.

To comprehend the variability of \( \hat{p} \) across many possible random samples from the same population, we look to the sampling distribution, which is where concepts like the mean and standard deviation of the sampling distribution come into play. The mean of the sampling distribution of \( \hat{p} \) is equal to 'p', and the standard deviation gives us an idea of how much \( \hat{p} \) would typically vary from sample to sample.

Normality Condition

Normality condition is one of the assumptions we check to ensure the validity of conclusions drawn using the normal model. In the context of sampling distributions of proportions, the normality condition requires that the values of \( np \) and \( n(1-p) \) both be greater than 10, which can also be understood as having at least 10 expected successes and 10 expected failures in our sample.

Let's break down this rule with our social network users' example. We calculate \( np = 0.56 \times 50 = 28 \) and \( n(1-p) = 50 \times 0.44 = 22 \) for a sample of 50 users. Since both of these products are greater than 10, the normality condition is satisfied, affirming that the sampling distribution of \( \hat{p} \) is approximately normal. If this condition were not met, the use of normal approximation methods to solve questions about the sampling distribution might not be appropriate, which could lead to inaccurate inferences.

Ensuring the normality condition allows us to proceed confidently with using the normal model to create confidence intervals or conduct hypothesis tests about the proportion. It is an essential step in data analysis that helps guarantee the reliability of our statistical conclusions.