Questions: For samples of the specified size from the population described, find the mean and standard deviation of the sample mean x. One barge from Inland Waterways, Inc. can carry a load of 6284.8 lb. Records of past trips show that the weights of the cans that it carries have a mean of 94 lb and a standard deviation of 16 lb. For samples of size 64, find the mean and standard deviation of x̄.

Solution Steps

The mean of the sample mean \( \bar{x} \) is equal to the population mean. Given that the population mean is \( \mu = 94 \), we have:

\[ \bar{x} = \mu = 94 \]

The standard deviation of the sample mean \( \bar{x} \) is calculated using the formula:

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

where:

• \( \sigma \) is the population standard deviation, which is \( 16 \),
• \( n \) is the sample size, which is \( 64 \).

Substituting the values, we get:

\[ \sigma_{\bar{x}} = \frac{16}{\sqrt{64}} = \frac{16}{8} = 2 \]

Final Answer