According to a student project for statistics 250, the amount of money spent on texts per semester by the population of students at a small liberal arts college is normally distributed with a mean of $426 and a standard deviation of $54.

Determine the percent of all samples of 4 students at the college that spent mean amount of money within $40.50 of the population mean amount.

To determine the percent of all samples of 4 students at the college that spent a mean amount of money within $40.50 of the population mean amount, we can use the concept of the sampling distribution of the sample mean.

Step 1: Calculate the standard deviation of the sample mean (also known as the standard error).

The standard deviation of the sample mean, denoted as σx̅, can be calculated using the formula:

σx̅ = σ / sqrt(n)

where σ is the standard deviation of the population (given as $54) and n is the sample size (which is 4 in this case).

σx̅ = $54 / sqrt(4) = $54 / 2 = $27

Step 2: Calculate the z-score for the given range.

To determine the percent of samples within a specific range, we need to calculate the z-scores corresponding to the lower and upper bounds of that range.

First, we calculate the z-score for the lower bound of $40.50 below the population mean:

Lower bound: $426 - $40.50 = $385.50

z1 = (x1 - μ) / σx̅ = ($385.50 - $426) / $27 = -1.50

Next, we calculate the z-score for the upper bound of $40.50 above the population mean:

Upper bound: $426 + $40.50 = $466.50

z2 = (x2 - μ) / σx̅ = ($466.50 - $426) / $27 = 1.50

Step 3: Calculate the area under the standard normal curve between the two z-scores.

Using a z-table or a calculator that can calculate areas under the normal curve, we find that the area between z = -1.50 and z = 1.50 is approximately 0.8664.

Step 4: Convert the area to a percentage.

Since we want to determine the percent of samples within this range, we multiply the probability by 100:

Percent = 0.8664 * 100 = 86.64%

Therefore, approximately 86.64% of all samples of 4 students at the college spent a mean amount of money within $40.50 of the population mean amount.