Question

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 (c).

(a) What is the probability a randomly selected student in the city will read more than 96 words per minute? The probability is [Enter the probability rounded to four decimal places]. Interpret this probability.

Select the correct choice below and fill in the answer box within your choice:

A. If 100 different students were chosen from this population, we would expect [Enter expected outcome] to read less than 96 words per minute.

B. If 100 different students were chosen from this population, we would expect [Enter expected outcome] to read more than 96 words per minute.

C. If 100 different students were chosen from this population, we would expect [Enter expected outcome] to read exactly 96 words per minute.

(b) What is the probability that a random sample of 11 second-grade students from the city results in a mean reading rate of more than 96 words per minute? The probability is [Enter the probability rounded to four decimal places]. Interpret this probability.

Select the correct choice below and fill in the answer box within your choice:

A. If 100 independent samples of n = 11 students were chosen from this population, we would expect [Enter expected outcome].

B. If 100 independent samples of n = 11 students were chosen from this population, we would expect [Enter expected outcome].

C. If 100 independent samples of n = 11 students were chosen from this population, we would expect [Enter expected outcome].

(c) What is the probability that a random sample of 22 second-grade students from the city results in a mean reading rate of more than 96 words per minute? The probability is [Enter the probability rounded to four decimal places]. Interpret this probability.

Select the correct choice below and fill in the answer box within your choice:

A. If 100 independent samples of n = 22 students were chosen from this population, we would expect [Enter expected outcome].

B. If 100 independent samples of n = 22 students were chosen from this population, we would expect [Enter expected outcome].

C. If 100 independent samples of n = 22 students were chosen from this population, we would expect [Enter expected outcome].

Answer 1

The probabilities of a randomly selected student reading more than 96 words per minute, and the probabilities for sample means of 11 and 22 students exceeding the same rate have been found using the standard normal distribution and the central limit theorem. The expected number of students or samples exceeding 96 wpm for parts (a), (b), and (c) have been provided accordingly.

Step-by-step explanation:

To solve this problem, we'll first use the standard normal distribution to find the probability that a randomly selected student reads more than 96 words per minute. We'll then apply the central limit theorem to find the probabilities for the scenarios where the means of samples of 11 and 22 students have to be calculated.

Part (a)

The z-score for a reading speed of 96 wpm is calculated as follows:

Z = (X - μ) / σ = (96 - 91) / 10 = 0.5

Using the standard normal distribution, we look up the z-score of 0.5 to find the probability that a student reads more than 96 wpm. This is equivalent to 1 minus the cumulative probability at z = 0.5.

Bold: The probability that a randomly selected student will read more than 96 words per minute is 0.3085. So, if 100 different students were chosen from this population, we would expect 31 to read more than 96 words per minute (B).

Part (b)

For a sample of 11 students, the sampling distribution of the sample mean will have a standard deviation equal to the population standard deviation divided by the square root of the sample size (σ/√n). Calculating the z-score for the sample mean of more than 96 words per minute and then finding the corresponding probability will give us:

Bold: The probability for a sample of 11 students to have a mean reading rate of more than 96 words per minute is 0.1573. Therefore, if 100 independent samples of n = 11 students were chosen, we would expect 16 to have a mean reading rate of more than 96 wpm (A).

Part (c)

Applying the same approach as in part (b), but with a sample size of 22, the probability will naturally increase since the standard deviation of the sampling distribution decreases with the larger sample size.

Bold: The probability that a sample of 22 students has a mean reading rate of more than 96 words per minute is 0.2262. In the case of 100 independent samples of n = 22 students, we would expect 23 to have a mean reading rate of more than 96 wpm (A).