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 0.4723
(Round to four decimal places as needed.)
(c) Find the probability that a randomly selected high school student who took the reading portion of the test has a score that is more than 34.9 .
The probability of a student scoring more than 34.9 is
(Round to four decimal places as needed.)
Answer
Final Answer: 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 \(\boxed{0.3498}\). 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 \(\boxed{0.4723}\). The probability that a randomly selected high school student who took the reading portion of the test has a score that is more than 34.9 is approximately \(\boxed{0.0201}\).
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, between 19.6 and 26.8, and more than 34.9.
Step 3 ：These probabilities can be found using the z-score formula and the standard normal distribution table. The z-score formula is \(z = \frac{X - \mu}{\sigma}\), where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 4 ：For part (a), we need to find the z-score for a score of 21 and look up the corresponding probability in the standard normal distribution table. The z-score is calculated as \(z = \frac{21 - 23.2}{5.7} = -0.386\). The corresponding probability is approximately 0.3498.
Step 5 ：For part (b), we need to find the z-scores for scores of 19.6 and 26.8, look up the corresponding probabilities in the standard normal distribution table, and subtract the smaller probability from the larger one. The z-scores are calculated as \(z_1 = \frac{19.6 - 23.2}{5.7} = -0.632\) and \(z_2 = \frac{26.8 - 23.2}{5.7} = 0.632\). The corresponding probabilities are subtracted to give approximately 0.4723.
Step 6 ：For part (c), we need to find the z-score for a score of 34.9, look up the corresponding probability in the standard normal distribution table, and subtract this probability from 1, since we want the probability of a score being more than 34.9. The z-score is calculated as \(z = \frac{34.9 - 23.2}{5.7} = 2.053\). The corresponding probability is subtracted from 1 to give approximately 0.0201.
Step 7 ：Final Answer: 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 \(\boxed{0.3498}\). 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 \(\boxed{0.4723}\). The probability that a randomly selected high school student who took the reading portion of the test has a score that is more than 34.9 is approximately \(\boxed{0.0201}\).
