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.