Question

Suppose that a sample of size 100 is to be drawn from a population with standard deviation 10.

Answer 1

Answer:

a) 68% probability that the sample mean will be within 1 of the value of μ.

b)

1)

Approximately 95% of the time, x will be within 2 of μ.

2)

Approximately 0.3% of the time, x will be farther than 3 from μ.

The last:

n = 40

n = 65

n = 130

n = 520

Step-by-step explanation:

To solve this problem, it is important to know two concepts: The Empirical Rule and the Central Limit Theorem.

Empirical Rule

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

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

95% of the measuers are within 2 standard deviations 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 $\mu$ and standard deviation $\sigma$, a large sample size, of at least 30, can be approximated to a normal distribution with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$.

Suppose that a sample of size 100 is to be drawn from a population with standard deviation 10.

So $\sigma = 10, n = 100, s = \frac{10}{\sqrt{100}} = 1$

(a) What is the probability that the sample mean will be within 1 of the value of μ?

Within 1 is within one standard deviation of the mean $\mu$.

So there is a 68% probability that the sample mean will be within 1 of the value of μ.

(b) For this example (n = 100, σ = 10), complete each of the following statements by computing the appropriate value. (Round the answers to the nearest whole number.)

(1) Approximately 95% of the time, x will be within___ of μ.

By the empirical rule, 95% of the measures are within 2 standard deviations of the mean. In our sample, the standard deviation is 1.

So

Approximately 95% of the time, x will be within 2 of μ.

(2) Approximately 0.3% of the time, x will be farther than___ from μ.

By the empirical rule, 99.7% of the measures are within 3 standard deviations of the mean. In the other 0.3% of the time, the measures are farther than 3 standard deviations of the mean. In our sample, the standard deviation is 1.

So:

Approximately 0.3% of the time, x will be farther than 3 from μ.

A random sample is selected from a population with mean μ = 100 and standard deviation σ = 10. For which of the sample sizes would it be reasonable to think that the xsampling distribution is approximately normal in shape? (Select all that apply.)

As we saw above, in the central Limit theorem, we should use a sample size of at least 30. So

n = 40

n = 65

n = 130

n = 520