A population has a mean of 200 and a standard deviation of 50. Suppose a sample of size 100 is selected and x is used to estimate m. a.What is the probability that the sample mean will be within ±5 of the population mean?b. What is the probability that the sample mean will be within ±10 of the population mean?

a) 68% probability that the sample mean will be within ±5 of the population mean.

b) 95% probability that the sample mean will be within ±10 of the population mean.

Step-by-step explanation:

To solve this problem, we have to understand the Empirical Rule and the Central Limit Theorem

Empirical Rule

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

Mean:

Standard deviation of the population:

Size of the sample:

Standard deviation for the sample mean:

a.What is the probability that the sample mean will be within ±5 of the population mean?

5 is one standard deviation from the mean.

By the Empirical Rule, 68% probability that the sample mean will be within ±5 of the population mean.

b.What is the probability that the sample mean will be within ±10 of the population mean?

10 is two standard deviations from the mean

By the Empirical Rule, 95% probability that the sample mean will be within ±10 of the population mean.

a orange at the grocery store idk what other product that could be