Suppose that the number of credit hours a student takes follows a normal distribution with a mean of 17 credit hours and a standard deviation of 2 credit hours. A normal course load is between 12 and 19 hours. What proportion of students have a normal course load?

Question

BlackTom AI · Accepted Answer

We start by identifying the distribution details: the number of credit hours is modeled as a normal random variable with mean μ = 17 and standard deviation σ = 2. A normal course load is defined as between 12 and 19 hours. We want P(12 ≤ X ≤ 19).

First, standardize the bounds to Z ~ N(0,1):
- For the lower bound 12: z = (12 − 17)/2 = -5/2 = -2.5.
- For the upper bound 19: z = (19 − 17)/2 = 2/2 = 1.
Thus we seek P(-2.5 ≤ Z ≤ 1).

Using the standard normal CDF Φ:
P(-2.5 ≤ Z ≤ 1) = Φ(1) − Φ(-2.5).
From standard normal tables (or a calculator), Φ(1) ≈ 0.8413 and Φ(-2.5) ≈ 0.0062 (since Φ(-z) = 1 − Φ(z) and Φ(2.5) ≈ 0.9938).
Therefore, P ≈ 0.8413 − 0.0062 = 0.8351.

This proportion, about 0.835, lies between 0.5 and 0.9 when expressed as a decimal between 0 and 1. Now evaluate each answer option:

Option 1: 'between 0.5 and 0.9' — This aligns with the calculated value of roughly 0.835, so it is consistent. It correctly captures the magnitude of the probability.

Option 2: 'less than 0.05' — This would imply a probability under 5%. Our computed value is well above 5%, so this is incorrect.

Option 3: 'greater than 0.9' — This would require a probability above 90%, which is not supported by the calculation; 0.835 is clearly less than 0.9, so this is incorrect.

Option 4: 'between 0.05 and 0.49' — This interval would place the probability between 5% and 49%. Our value 83.5% does not fall in this range, so this is incorrect.

In summary, the only option that matches the computed probability is the first one, which states the proportion is between 0.5 and 0.9. The other choices mischaracterize the magnitude of the probability based on the standard-normal calculation.

Suppose that the number of credit hours a student takes follows a normal distribution with a mean of 17 credit hours and a standard deviation of 2 credit hours. A normal course load is between 12 and 19 hours. What proportion of students have a normal course load?单项选择题

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