Problem 17 Question: The reading speed of sixth-grade students is approximately normal, with a mean speed of 124 words per minute and a standard deviation of 24 words per minute. Find the probability that a randomly selected sixth-grade student reads a. less than 100 words per minute. \& Answer: b. more than 140 words per minute. o Answer: c. between 110 and 130 words per minute. a Answer:

To find the probability that a randomly selected sixth-grade student reads less than 100 words per minute, we can use the z-score formula: \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value we are interested in, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. In this case, \( X = 100 \), \( \mu = 124 \), and \( \sigma = 24 \). Calculating the z-score: \( z = \frac{(100 - 124)}{24} \approx -1.00 \). Next, we can refer to the standard normal distribution table, or use a calculator, to find the probability associated with \( z = -1.00 \), which is approximately 0.1587. This means there’s about a 15.87% chance that a sixth grader reads less than 100 words per minute. Now for part b, we’re looking for the probability that a student reads more than 140 words per minute. First, we’ll calculate the z-score for 140: \( z = \frac{(140 - 124)}{24} \approx 0.67 \). Now, checking the standard normal distribution table, the area to the left of \( z = 0.67 \) is approximately 0.7486. Therefore, the probability of reading more than 140 words per minute is: \( P(X > 140) = 1 - 0.7486 \approx 0.2514 \). That means there’s about a 25.14% chance that a randomly selected sixth grader reads faster than 140 words per minute. Finally, for part c, we'll find the probability that a student reads between 110 and 130 words per minute. Calculate the z-scores for both values: For 110: \( z = \frac{(110 - 124)}{24} \approx -0.58 \). For 130: \( z = \frac{(130 - 124)}{24} \approx 0.25 \). Using the standard normal table (or calculator), the area for \( z = -0.58 \) is approximately 0.2810, and for \( z = 0.25 \), it is approximately 0.5987. To find the probability between these two values, we subtract: \( P(110 < X < 130) = P(X < 130) - P(X < 110) \approx 0.5987 - 0.2810 \approx 0.3177 \). Thus, there’s about a 31.77% chance that a sixth grader reads between 110 and 130 words per minute!