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A population has a mean of 200 and a standard deviation of 60. Suppose a sample of size 100 is selected and is used to estimate . Use z-table.

A) What is the probability that the sample mean will be within +/- 7 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.)

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Final answer:

To find the probability that the sample mean will be within +/- 7 of the population mean, we need to calculate the standard error of the mean and use the z-table to find the probability.

Step-by-step explanation:

To find the probability that the sample mean will be within +/- 7 of the population mean, we need to calculate the standard error of the mean and then use the z-table to find the probability. The formula for the standard error of the mean is:

SE = σ / √n

where σ is the population standard deviation and n is the sample size. In this case, σ = 60 and n = 100. Plugging in these values, we get:

SE = 60 / √100 = 60 / 10 = 6

Now, we can calculate the z-scores for +/- 7 using the formula:

z = (x - μ) / SE

where x is the value we want to find the z-score for, μ is the population mean, and SE is the standard error of the mean. Plugging in the values for +/- 7, we get:

z1 = (200 + 7 - 200) / 6 = 7 / 6 = 1.17

z2 = (200 - 7 - 200) / 6 = -7 / 6 = -1.17

Using the z-table, we can find the probabilities associated with these z-scores. The probability that the sample mean will be within +/- 7 of the population mean is the sum of these probabilities, since the area under the curve between z1 and z2 represents the desired probability. Look up the z-scores in the z-table and sum the corresponding probabilities:

P(z < 1.17) = 0.8790

P(z < -1.17) = 0.1209

Summing these probabilities, we get:

P(-1.17 < z < 1.17) = P(z < 1.17) - P(z < -1.17) = 0.8790 - 0.1209 = 0.7581

Therefore, the probability that the sample mean will be within +/- 7 of the population mean is approximately 0.7581.

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