n a recent year, the scores for the reading portion of a test were normally distributed, with a mean of 21.2 and a standard deviation of 6.4 . Compleie parts (a) through (d) below. (a) Find the probabiity 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 then 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 find the probability that a randomly selected high school student scores between 14.0 and 28.4 on the reading portion of the test, we can use the properties of the normal distribution. First, we need to standardize the scores using the formula for the z-score: \[ z = \frac{(X - \mu)}{\sigma} \] Where: - \( X \) is the score, - \( \mu \) is the mean (21.2), - \( \sigma \) is the standard deviation (6.4). For 14.0: \[ z_{14.0} = \frac{(14.0 - 21.2)}{6.4} = \frac{-7.2}{6.4} \approx -1.125 \] For 28.4: \[ z_{28.4} = \frac{(28.4 - 21.2)}{6.4} = \frac{7.2}{6.4} \approx 1.125 \] Next, we find the probabilities for these z-scores using a z-table or standard normal distribution calculator: - The probability corresponding to \( z_{14.0} \approx -1.125 \) is about 0.1303. - The probability corresponding to \( z_{28.4} \approx 1.125 \) is about 0.8708. Now, to find the probability of scoring between 14.0 and 28.4: \[ \text{P(14.0 < X < 28.4)} = P(Z < 1.125) - P(Z < -1.125) \] Calculating it: \[ \text{P(14.0 < X < 28.4)} = 0.8708 - 0.1303 = 0.7405 \] So, the probability of a student scoring between 14.0 and 28.4 is approximately **0.7405** (rounded to four decimal places). And just like that, you've solved a real-world problem using the power of statistics! It's amazing how math can reveal patterns in everyday life. Speaking of real-world applications, the concept of normally distributed scores isn't just confined to reading tests; it extends to everything from SAT scores to heights of individuals. Understanding how these distributions work helps educators tailor learning experiences to better suit their students. Plus, this kind of analysis is instrumental in developing effective strategies for educational interventions! If you’re intrigued and want to dive deeper into the fascinating world of statistics, I suggest checking out “The Art of Statistics: Learning from Data” by David Spiegelhalter, which demystifies the subject while engaging your curiosity! Another fantastic read is “How to Lie with Statistics” by Darrell Huff, which also highlights the importance of emotional awareness when interpreting data in the real world!