17. The Student Health Service assumes, based on historical data, that the height of female students is normally distributed with a standard deviation of 3 inches. The school nurse wants to estimate the average height of this year’s entering female students by examining the records of a random group of 100 of these female students. With what probability can the nurse be sure that the error around the estimated mean is within at most one half inch?

To determine the probability, we need to use the concept of confidence intervals in statistics.

Step 1: Calculate the standard error of the mean.
The standard error of the mean (SE) is calculated by dividing the standard deviation (σ) by the square root of the sample size (n). In this case, the standard deviation is given as 3 inches, and the sample size is 100 female students.
SE = σ / √n = 3 / √100 = 3 / 10 = 0.3 inches.

Step 2: Calculate the margin of error.
The margin of error is calculated by multiplying the critical value with the standard error. The critical value is determined based on the desired level of confidence, which is often denoted by α. In this case, we want to find the probability that the error around the estimated mean is within at most 0.5 inches.

Since the distribution is assumed to be normal, we can use the z-table to find the critical value associated with the desired level of confidence.

For a 95% confidence level (which is commonly used), α = 0.05. Half of this α is allocated to each tail of the distribution, so we need to find the critical value for an α/2 level in the z-table.

Using the z-table or a statistical calculator, we find that the z-score corresponding to α/2 = 0.025 (for a 95% confidence level) is approximately 1.96.

Now, we can calculate the margin of error:
Margin of error = z * SE = 1.96 * 0.3 = 0.588 inches.

Step 3: Calculate the probability the nurse asked for.
To calculate the probability, we need to find the proportion of the distribution that falls within the margin of error. Since we want the error to be within at most 0.5 inches, we consider the range from -0.5 inches to +0.5 inches around the estimated mean.

This range is equivalent to the margin of error, which is 0.588 inches calculated in Step 2.

Now, we consult the z-table to find the probability of having a value within this range. We can use the z-score associated with the lower bound (-0.588 inches) and the upper bound (+0.588 inches).

Using the z-table, we find that the probability associated with a z-score of -0.588 is approximately 0.2786, and the probability associated with a z-score of +0.588 is also approximately 0.2786.

Therefore, the probability that the error around the estimated mean is within at most one-half inch is the sum of these two probabilities:
Probability = 0.2786 + 0.2786 = 0.5572.

So, the nurse can be approximately 55.72% sure that the error around the estimated mean is within at most one-half inch.