Suppose a random sample of n = 16 observations is selected from a population tha

Question

Suppose a random sample of n = 16 observations is selected from a population that is normally distributed with mean equal to 109 and standard deviation equal to 11.

(a) Give the mean and the standard deviation of the sampling distribution of the sample mean x. (Enter your standard deviation to two decimal places.)

(b) Find the probability that x exceeds 114. (Round your answer to four decimal places.)

(c) Find the probability that the sample mean deviates from the population mean = 109 by no more than 3. (Round your answer to four decimal places.)

Explanation / Answer

A) the mean of the sampling distribution is the same as the population mean: 109
The standard deviation of the sampling distribution is called the standard error and is equal to s/ n = 11/ 16 = 2.75

Use a standard normal table to find the following probabilities:

B) P(xbar > 114)
= P((xbar-µ)/(s/ n) > (114-109)/(11/ 16))
= P(z > 1.818)
= 1P ( Z<1.818 )
= 10.9656=0.0344

C) P(106 < xbar < 112)
= P((106-109)/(11/ 16) < (xbar-µ)/(s/ n) < (112-109)/(11/ 16))
= P(-1.09 < z < 1.09) = 0.7242

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