The daily revenue X at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is 4400 and the variance is 400. The distribut... The daily revenue X at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is 4400 and the variance is 400. The distribution is skewed to the right due to several high volume days (football game days). Suppose that 150 days are randomly selected and the average daily revenue is computed. Which of the following describes the sampling distribution of the sample mean?
Understand the Problem
The question describes a scenario involving the daily revenue at a university snack bar. We are given the population mean and variance. We are asked to describe the sampling distribution of the sample mean when 150 days are randomly selected. The key concept here is the Central Limit Theorem, which states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, if the sample size is large enough. Since the sample size is 150, we can consider it large enough to apply the Central Limit Theorem.
Answer
The sampling distribution of the sample mean $\overline{x}$ is approximately normal with a mean of $480$ and a standard deviation of $\frac{70}{\sqrt{150}} \approx 5.715$.
Answer for screen readers
The sampling distribution of the sample mean $\overline{x}$ is approximately normal with a mean $\mu_{\overline{x}} = 480$ and a standard deviation $\sigma_{\overline{x}} = \frac{70}{\sqrt{150}} \approx 5.715$. We can write this as $\overline{x} \sim N(480, (5.715)^2)$.
Steps to Solve
- Identify the parameters of the population.
The population mean is $\mu = 480$ and the population variance is $\sigma^2 = 4900$. This means the population standard deviation is $\sigma = \sqrt{4900} = 70$.
- Apply the Central Limit Theorem.
Since the sample size $n = 150$ is large (generally, $n > 30$ is considered large enough), the sampling distribution of the sample mean $\overline{x}$ will be approximately normal, regardless of the shape of the population distribution.
- Determine the mean of the sampling distribution.
The mean of the sampling distribution of the sample mean, denoted as $\mu_{\overline{x}}$, is equal to the population mean $\mu$. Therefore, $\mu_{\overline{x}} = \mu = 480$.
- Determine the standard deviation (standard error) of the sampling distribution.
The standard deviation of the sampling distribution of the sample mean, also known as the standard error, is given by $\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. In this case, $\sigma_{\overline{x}} = \frac{70}{\sqrt{150}} \approx \frac{70}{12.247} \approx 5.715$.
- Describe the sampling distribution.
The sampling distribution of the sample mean $\overline{x}$ is approximately normal with a mean of 480 and a standard deviation of approximately 5.715. We express this as $\overline{x} \sim N(480, (5.715)^2)$.
The sampling distribution of the sample mean $\overline{x}$ is approximately normal with a mean $\mu_{\overline{x}} = 480$ and a standard deviation $\sigma_{\overline{x}} = \frac{70}{\sqrt{150}} \approx 5.715$. We can write this as $\overline{x} \sim N(480, (5.715)^2)$.
More Information
The Central Limit Theorem is a cornerstone of statistics, allowing us to make inferences about population parameters based on sample statistics. In this case, even if we don't know the shape of the distribution of daily snack bar revenue, we know that the distribution of sample means will be approximately normal, allowing us to calculate probabilities and confidence intervals related to the true population mean.
Tips
A common mistake is forgetting to divide the population standard deviation by the square root of the sample size when calculating the standard deviation of the sampling distribution (standard error). Another common mistake is assuming that the sampling distribution is normal regardless of the sample size. The Central Limit Theorem applies when the sample size is sufficiently large (typically $n > 30$).
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