Math / Data & Statistics

QuestionStandard 15 Suppose the mean commute time among all NKU students is 27.3 minutes with a standard deviation of 9.38 minutes. Consider samples of 49 NKU students for which the sample mean is calculated. A. Fully describe the sampling distribution of the sample mean.

Studdy Solution

STEP 1

1. The population mean commute time is $\mu = 27.3$ minutes.
2. The population standard deviation is $\sigma = 9.38$ minutes.
3. The sample size is $n = 49$ .
4. The Central Limit Theorem applies because the sample size is large ( $n \geq 30$ ).

STEP 2

1. Identify the mean of the sampling distribution.
2. Calculate the standard deviation of the sampling distribution (standard error).
3. Describe the shape of the sampling distribution.

STEP 3

The mean of the sampling distribution of the sample mean is equal to the population mean:
$\mu_{\bar{x}} = \mu = 27.3$

STEP 4

Calculate the standard error of the mean, which is the standard deviation of the sampling distribution:
$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{9.38}{\sqrt{49}}$
$\sigma_{\bar{x}} = \frac{9.38}{7}$
$\sigma_{\bar{x}} = 1.34$

STEP 5

According to the Central Limit Theorem, the shape of the sampling distribution of the sample mean will be approximately normal because the sample size is large ( $n = 49$ ).
The sampling distribution of the sample mean is fully described as follows: - Mean: $27.3$ - Standard Error: $1.34$ - Shape: Approximately normal

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