Answer to Question #98237 in Statistics and Probability for Alexus

Question #98237
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.
1
Expert's answer
2019-11-08T14:20:39-0500

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.



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Comments

Assignment Expert
15.07.21, 22:30

Dear Fhely Montefalcon, please use the panel for submitting a new question.


Fhely Montefalcon
14.06.21, 06:47

Mean is 83 and sample size is 39

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