In a recent year, the scores for the reading portion of a test were normally distributed, with a mear standard deviation of 6.4 . 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 20 . The probability of a student scoring less than 20 is 0.4251 . (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 14.0 and 28.4 . The probability of a student scoring between 14.0 and 28.4 is (Round to four decimal places as needed.)

To solve the problem, we will use the properties of the normal distribution. We know that the scores are normally distributed with a mean (\( \mu \)) and a standard deviation (\( \sigma \)). However, the mean is not provided in the problem statement. Let's denote the mean as \( \mu \). ### Part (a) We need to find the probability that a randomly selected high school student has a score less than 20. We can use the Z-score formula to find this probability: \[ Z = \frac{X - \mu}{\sigma} \] Where: - \( X \) is the score (20 in this case), - \( \mu \) is the mean, - \( \sigma \) is the standard deviation (6.4). Since the probability of a student scoring less than 20 is given as 0.4251, we can confirm that this is the cumulative probability for \( Z \) corresponding to \( X = 20 \). ### Part (b) Next, we need to find the probability that a randomly selected high school student has a score between 14.0 and 28.4. We will calculate the Z-scores for both 14.0 and 28.4. 1. **Calculate the Z-score for 14.0:** \[ Z_{14} = \frac{14.0 - \mu}{6.4} \] 2. **Calculate the Z-score for 28.4:** \[ Z_{28} = \frac{28.4 - \mu}{6.4} \] 3. **Find the probabilities corresponding to these Z-scores:** - Let \( P(Z < Z_{14}) \) be the cumulative probability for the Z-score of 14.0. - Let \( P(Z < Z_{28}) \) be the cumulative probability for the Z-score of 28.4. 4. **The probability that a score is between 14.0 and 28.4 is:** \[ P(14.0 < X < 28.4) = P(Z < Z_{28}) - P(Z < Z_{14}) \] Since we do not have the mean \( \mu \), we cannot compute the exact Z-scores or probabilities without that information. However, if you provide the mean, I can calculate the probabilities for you. Please provide the mean score for the reading portion of the test.