Answer to Question #98237 in Statistics and Probability for Alexus

Question

Suppose x has a distribution with a mean of 60 and a standard deviation of 27. Random samples of size n = 36 are drawn.
(a) Describe the x bar distribution.
x bar has an unknown distribution.
x bar has an approximately normal distribution.
x bar has a Poisson distribution.
x bar has a geometric distribution.
x bar has a normal distribution.
x bar has a binomial distribution.
Correct: Your answer is correct.
Compute the mean and standard deviation of the distribution. (For each answer, enter a number.)

mu sub x bar = mu sub x bar =

sigma sub x bar = sigma sub x bar =

(b) Find the z value corresponding to x bar = 69. (Enter an exact number.)
z =
Incorrect: Your answer is incorrect.


(c) Find P(x bar < 69). (Enter a number. Round your answer to four decimal places.)
P(x bar < 69) = P(x bar < 69)

(d) Would it be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 69? Explain.

Answer 1

The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30. 

a) Since random samples of size n = 36 (36>30) are drawn, then $\bar{X}$  has an approximately normal distribution.

$\mu_{\bar{X}}=mean=60$

$\sigma_{\bar{X}}=\sigma/\sqrt{n}=27/\sqrt{36}=4.5$

$\bar{X}\sim N(60, 4.5^2)$

Then

$Z={\bar{X}-\mu_{\bar{X}} \over \sigma_{\bar{X}}}\sim N(0, 1)$

b)

If $\bar{X}=69,$ then

$Z={69-60 \over 4.5}=2$

c)

$P(\bar{X}<69)=P(Z<2)\approx0.9772$

d)

$\mu_{\bar{X}}-2\sigma_{\bar{X}}=60-2(4.5)=51$

$\mu_{\bar{X}}+2\sigma_{\bar{X}}=60+2(4.5)=69$

Maximum usual value is $69.$ Minimum usual value is $51.$

It would be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 51.

It would be unusual for a random sample of size 36 from the x distribution to have a sample mean greater than 69.

It would not be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 69.