Question

Assume that cans are filled so that the actual amounts have a mean of 14.00 ounces. A random sample of 36 cans has a mean amount of 14.36 ounces. The distribution of sample means of size 36 is normal with an assumed mean of 14.00 ounces and a standard deviation of 0.04 ounces. How many standard deviations is the sample mean from the mean of the distribution of sample means

A) 0.36 standard deviations
B) 0.67 standard deviations
C) 1.00 standard deviation
D) 54 standard deviations

Answer 1

B) 0.67 standard deviations

Step-by-step explanation:

The formula to find the number of standard deviations the sample mean is from the mean of the distribution of sample means is (sample mean - population mean) / (standard deviation of sample means / square root of sample size). Substituting the values given: (14.36 - 14.00) / (0.04 / √36) = 0.36 / (0.04 / 6) = 0.36 / 0.0067 ≈ 0.67. This indicates that the sample mean of 14.36 ounces is 0.67 standard deviations from the mean of the distribution of sample means.

The calculation involves finding the difference between the sample mean and the population mean, divided by the standard deviation of sample means (which is the standard deviation of the population divided by the square root of the sample size). In this case, the calculation results in approximately 0.67 standard deviations away from the mean of the distribution of sample means. This signifies that the sample mean of 14.36 ounces is 0.67 standard deviations from the expected mean of 14.00 ounces for the distribution of sample means. Understanding these standard deviations helps in assessing the variability and significance of the sample mean within the distribution of sample means.