Questions: For people under 50, the level of glucose in the blood (in milligrams per deciliter of blood) after a 12-hour fast have a standard deviation of 25 and a mean of μ. What is the probability that, for a sample of size 49 readings, the sample mean is within 7 of μ?

Solution Steps

To solve this problem, we need to use the concept of the sampling distribution of the sample mean. The standard deviation of the sample mean (also known as the standard error) is the population standard deviation divided by the square root of the sample size. We can then use the standard normal distribution to find the probability that the sample mean is within a certain range of the population mean.

The standard error of the sample mean is calculated using the formula:

\[ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} \]

where \(\sigma = 25\) is the population standard deviation and \(n = 49\) is the sample size. Substituting the given values:

\[ \text{Standard Error} = \frac{25}{\sqrt{49}} = 3.5714 \]

The Z-score for the margin of error is calculated using:

\[ Z = \frac{\text{Margin}}{\text{Standard Error}} \]

Given that the margin is 7:

\[ Z = \frac{7}{3.5714} = 1.9600 \]

The probability that the sample mean is within 7 units of the population mean is found using the cumulative distribution function (CDF) of the standard normal distribution. The probability is given by:

\[ P(-Z < Z < Z) = \text{CDF}(Z) - \text{CDF}(-Z) \]

Substituting the Z-score:

\[ P(-1.9600 < Z < 1.9600) = 0.9500 \]

Final Answer