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 the mean amount of money within $40.50 of the population mean amount, we need to calculate the z-score and then find the corresponding percentage using a standard normal distribution table.
The formula to calculate the z-score is:
z = (x - μ) / σ
where:
- z is the z-score
- x is the value we want to find the probability for
- μ is the mean of the population
- σ is the standard deviation of the population
In this case, we want to find the probability for 4 students who spent the mean amount within $40.50 of the population mean. Let's denote the mean amount spent by the sample of 4 students as x̄ (x-bar).
First, we need to calculate the z-scores for the lower and upper limits of $40.50:
Lower Limit: z1 = ($40.50 - μ) / σ
Upper Limit: z2 = ($40.50 + μ) / σ
Next, we need to find the area under the standard normal distribution curve between these two z-scores. We can use a standard normal distribution table or a statistical calculator to find this area.
Finally, we can find the probability by subtracting the area obtained in the previous step from 1 (since we want the probability that the mean amount spent by the sample is within the specified range).
Please note that the population mean and standard deviation in this case are given as $426 and $54, respectively.
Let's now calculate the z-scores:
Lower Limit: z1 = ($40.50 - $426) / $54
Upper Limit: z2 = ($40.50 + $426) / $54
z1 = -7.25
z2 = 8.44
Now we need to find the area between these two z-scores using a standard normal distribution table or a statistical calculator.
Using a standard normal distribution table or a statistical calculator, we find that the area between z1 and z2 is approximately 0.994.
Finally, we subtract this value from 1 to find the percent of all samples of 4 students at the college that spent the mean amount within $40.50 of the population mean amount:
Percent = (1 - 0.994) * 100
Percent ≈ 0.6
Therefore, approximately 0.6% of all samples of 4 students at the college would spend the mean amount within $40.50 of the population mean amount.