the times to pop a 3.4 - ounce bag of microwave popcorn without burning it are normally distributed with a…

the times to pop a 3.4 - ounce bag of microwave popcorn without burning it are normally distributed with a mean time of 140 seconds and a standard deviation of 20 seconds. a random sample of four bags is selected and the mean time to pop the bags is recorded. which of the following describes the sampling distribution of all possible samples of size four?\no approximately normal with a mean of 70 seconds and a standard deviation of 5 seconds\no approximately normal with a mean of 70 seconds and a standard deviation of 20 seconds\no approximately normal with a mean of 140 seconds and a standard deviation of 20 seconds\no approximately normal with a mean of 140 seconds and a standard deviation of 10 seconds

the times to pop a 3.4 - ounce bag of microwave popcorn without burning it are normally distributed with a mean time of 140 seconds and a standard deviation of 20 seconds. a random sample of four bags is selected and the mean time to pop the bags is recorded. which of the following describes the sampling distribution of all possible samples of size four?\no approximately normal with a mean of 70 seconds and a standard deviation of 5 seconds\no approximately normal with a mean of 70 seconds and a standard deviation of 20 seconds\no approximately normal with a mean of 140 seconds and a standard deviation of 20 seconds\no approximately normal with a mean of 140 seconds and a standard deviation of 10 seconds

Answer

Explanation:

Step1: Recall sampling - distribution properties

If the population is normally distributed with mean $\mu$ and standard deviation $\sigma$, the sampling - distribution of the sample mean $\bar{X}$ for samples of size $n$ is also normally distributed. The mean of the sampling - distribution of the sample mean $\mu_{\bar{X}}$ is equal to the population mean $\mu$, and the standard deviation of the sampling - distribution of the sample mean (also known as the standard error) $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$.

Step2: Identify population parameters

The population mean $\mu = 140$ seconds and the population standard deviation $\sigma = 20$ seconds, and the sample size $n = 4$.

Step3: Calculate the mean of the sampling - distribution of the sample mean

The mean of the sampling - distribution of the sample mean $\mu_{\bar{X}}=\mu=140$ seconds.

Step4: Calculate the standard deviation of the sampling - distribution of the sample mean

The standard deviation of the sampling - distribution of the sample mean $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\frac{20}{\sqrt{4}}=\frac{20}{2}=10$ seconds.

Answer:

Approximately Normal with a mean of 140 seconds and a standard deviation of 10 seconds.