In a recent year, the scores for the reading portion of a test were normally distributed, with a mean of 23.2 and a standard deviation of 5.7. Complete parts (a) through (d) below.
(a) Find the probability that a randomly selected high school student who took the reading portion of the test has a score that is less than 21.
The probability of a student scoring less than 21 is 0.3498 .
(Round to four decimal places as needed.)
(b) Find the probability that a randomly selected high school student who took the reading portion of the test has a score that is between 19.6 and 26.8 .
The probability of a student scoring between 19.6 and 26.8 is (Round to four decimal places as needed.)
Answer
Thus, the probability that a randomly selected high school student who took the reading portion of the test has a score that is between 19.6 and 26.8 is approximately \(\boxed{0.4723}\).
Step 1 ：Given that the scores for the reading portion of a test were normally distributed, with a mean of 23.2 and a standard deviation of 5.7.
Step 2 ：We are asked to find the probability that a randomly selected high school student who took the reading portion of the test has a score that is less than 21.
Step 3 ：We can calculate this probability using the properties of the normal distribution. The Z-score is a measure of how many standard deviations an element is from the mean.
Step 4 ：Calculate the Z-score for a score of 21 using the formula \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. This gives us a Z-score of approximately -0.386.
Step 5 ：We can then use the Z-table (standard normal distribution table) to find the probability associated with this Z-score. This gives us a probability of approximately 0.3498.
Step 6 ：Thus, the probability that a randomly selected high school student who took the reading portion of the test has a score that is less than 21 is approximately \(\boxed{0.3498}\).
Step 7 ：We are also asked to find the probability that a randomly selected high school student who took the reading portion of the test has a score that is between 19.6 and 26.8.
Step 8 ：We can calculate this probability by finding the Z-scores for 19.6 and 26.8, which are approximately -0.632 and 0.632, respectively.
Step 9 ：We can then use the Z-table to find the probabilities associated with these Z-scores. The difference between these probabilities gives us a probability of approximately 0.4723.
Step 10 ：Thus, the probability that a randomly selected high school student who took the reading portion of the test has a score that is between 19.6 and 26.8 is approximately \(\boxed{0.4723}\).
