The reading speed of second grade students in a large city is approximately normal, with a mean of 91 words per minute (wpm) and a standard deviation of 10 wpm. Complete parts (a) through (f). Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2). (b) What is the probability that a random sample of 10 second grade students from the city results in a mean reading rate of more than 97 words per minute? The probability is (Round to four decimal places as needed.) Interpret this probability. Select the correct choice below and fill in the answer box within your choice. A. If 100 different samples of $n=10$ students were chosen from this population, we would expect sample(s) to have a sample mean reading rate of exactly words per minute. B. If 100 different samples of $n=10$ students were chosen from this population, we would expect sample(s) to have a sample mean reading rate of less than words per minute. C. If 100 different samples of $n=10$ students were chosen from this population, we would expect sample(s) to have a sample mean reading rate of more thar 97 words per minute. (c) What is the probability that a random sample of 20 second grade students from the city results in a mean reading rate of more than 97 words per minute? The probability is
Solution
Step 1 ：The problem is asking for the probability that a random sample of 10 second grade students from the city results in a mean reading rate of more than 97 words per minute. This is a problem of normal distribution. We know that the mean reading speed is 91 wpm and the standard deviation is 10 wpm.
Step 2 ：We can use the z-score formula to calculate the z-score, which is \((X - μ) / (σ / √n)\), where X is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size.
Step 3 ：Substituting the given values into the formula, we get \(z = (97 - 91) / (10 / √10) = 1.8973665961010278\).
Step 4 ：We can then use the z-score to find the probability from the standard normal distribution table. The probability corresponding to this z-score is 0.028889785561798553.
Step 5 ：The probability that a random sample of 10 second grade students from the city results in a mean reading rate of more than 97 words per minute is approximately 0.0289. This means that if we randomly select 10 students, there is a 2.89% chance that their average reading speed will be more than 97 wpm.
Step 6 ：Final Answer: The probability is approximately \(\boxed{0.0289}\). If 100 different samples of \(n=10\) students were chosen from this population, we would expect approximately 2.89 samples to have a sample mean reading rate of more than 97 words per minute.
