Final answer:
The student loan debt for college graduates normally distributed with a mean of $25,700 and a standard deviation of $14,750 is used to find the probability of debt between specific ranges and the debt amounts for the middle 20% of the distribution.
Step-by-step explanation:
The question at hand involves calculating probabilities for a normally distributed variable, specifically relating to student loan debt for college graduates. The normal distribution of X, which represents the student loan debt of a randomly selected college graduate, is characterized by a mean (μ) of $25,700 and a standard deviation (σ) of $14,750.
B) To find the probability that a randomly selected graduate has a loan debt between $35,400 and $46,250, we utilize the properties of the normal distribution. The z-score for a value x is given by z = (x - μ) / σ. Once the z-scores for $35,400 and $46,250 are calculated, the areas under the normal curve can be obtained using a z-table or a statistical software. The difference between these two areas gives us the probability of a graduate having loan debt within that range.
C) To determine the loan debt amounts that fall in the middle 20% of the distribution, we look for z-scores that correspond to the 40th and 60th percentiles (since the middle 20% would span from the 40th to the 60th percentile). Using a z-table or statistical software, we identify the z-scores for these percentiles and then translate them back to dollar amounts using the formula x = (z * σ) + μ. These amounts represent the range of loan debts in the middle 20% of the distribution for college graduates.