Question

Ages of Proofreaders At a large publishing company, the mean age of proofreaders is 36.2 years and the standard deviation is 3.7 years. Assume the variable is normally distributed. Round intermediate z-value calculations to two decimal places and the final answers to at least four decimal places. Part 1 of 2 If a proofreader from the company is randomly selected, find the probability that his or her age will be between 36.5 and 38 years. Part 2 of 2 If a random sample of 15 proofreaders is selected, find the probability that the mean age of the proofreaders in the sample will be between 36.5 and 38 years. Assume that the sample is taken from a large population and the correction factor can be ignored.

Answer 1

The probability of a proofreader's age being between 36.5 and 38 years is 0.154. The probability of a sample mean age being between 36.5 and 38 years is 0.3484.

Step-by-step explanation:

To find the probability that a proofreader's age will be between 36.5 and 38 years, we need to calculate the z-scores for both ages and then find the area under the normal distribution curve between those z-scores.

Part 1: Calculating the z-scores
z1 = (36.5 - 36.2) / 3.7 = 0.08
z2 = (38 - 36.2) / 3.7 = 0.49

Using a standard normal distribution table or a calculator, we can find that the area to the left of z1 is 0.5328 and the area to the left of z2 is 0.6868. By subtracting these two values, we can find the probability that a proofreader's age will be between 36.5 and 38 years: 0.6868 - 0.5328 = 0.154.

Part 2: Calculating the probability for a sample mean
The mean of a sample of size 15 will still be 36.2 years, but the standard deviation will be calculated as the population standard deviation divided by the square root of the sample size: 3.7 / sqrt(15) = 0.9565. Now we can calculate the z-scores for the range of 36.5 to 38 years, using the new standard deviation: z1 = (36.5 - 36.2) / 0.9565 = 0.3134 and z2 = (38 - 36.2) / 0.9565 = 1.8789. We can find the probability of a sample mean being between 36.5 and 38 years by finding the area between these z-scores under the standard normal distribution curve. Using a standard normal distribution table or a calculator, the area to the left of z1 is 0.6198 and the area to the left of z2 is 0.9682. By subtracting these two values, we can find the probability: 0.9682 - 0.6198 = 0.3484.

Answer 2

The probability that a randomly selected proofreader's age will be between 36.5 and 38 years is 0.3421. The probability that the mean age of a random sample of 15 proofreaders will be between 36.5 and 38 years is 0.2649.

Part 1 of 2:

To find the probability that a randomly selected proofreader's age will be between 36.5 and 38 years, we need to calculate the z-scores for these values and use the standard normal distribution table. The z-score for 36.5 is calculated as follows:

z = (x - mean) / standard deviation = (36.5 - 36.2) / 3.7 = 0.08

The z-score for 38 is calculated as follows:

z = (x - mean) / standard deviation = (38 - 36.2) / 3.7 = 0.49

Using the standard normal distribution table, we can find the corresponding probabilities:

Probability (36.5 < x < 38) = Probability (0.08 < z < 0.49) = 0.1893 - 0.5328 = 0.3421

Therefore, the probability that a randomly selected proofreader's age will be between 36.5 and 38 years is 0.3421.

Part 2 of 2:

To find the probability that the mean age of a random sample of 15 proofreaders will be between 36.5 and 38 years, we use the Central Limit Theorem. The Central Limit Theorem states that sample means from a population with any distribution will tend to be approximately normally distributed as the sample size increases.

We can use the z-score formula for sample means:

z = (x - mean) / (standard deviation / sqrt(n))

Where x is the desired sample mean, mean is the population mean, standard deviation is the population standard deviation, and n is the sample size.

For 36.5, the z-score is calculated as:

z = (36.5 - 36.2) / (3.7 / sqrt(15)) = 0.31

For 38, the z-score is calculated as:

z = (38 - 36.2) / (3.7 / sqrt(15)) = 1.17

Using the standard normal distribution table, we can find the corresponding probabilities:

Probability (36.5 < x < 38) = Probability (0.31 < z < 1.17) = 0.3859 - 0.1210 = 0.2649

Therefore, the probability that the mean age of a random sample of 15 proofreaders will be between 36.5 and 38 years is 0.2649.