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# Normal Distribution, Mean and Standard Deviation

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7.40 A variable of a population has a mean of m = 35 and a standard deviation of o = 42.

a. If the variable is normally distributed, identify the sampling distribution of the sample for samples of 9.

b. Can you answer part (a) if the distribution of the variable under consideration is unknown? Explain your answer.

c. Can you answer part (a) if the distribution of the variable under consideration is unknown but the sample is 36 instead of 9? Why or why not?

7.42 A variable of a population has mean m and standard deviation o. For a large sample size n, answer the following questions.

a. Identify the distribution of x.
b. Does your answer to part (a) depend on n being large?
c. Identify the mean and the standard deviation of x.
d. Does your answer to part (c) depend on the sample size being large? Why or why not?

Use the 68.26-95.44-99.74 rule to answer the questions in parts (a)-(c).

7.54 A variable of a population is normally distributed with mean m and standard deviation o. For sample of size n, fill in the blanks. Justify your answers

a. 68.26% of all possible samples have means that lie within ________ of the population mean, m.
b. 95.44% of all possible samples have means that lie within _______ of the population mean, m.
c. 99.74% of all possible samples have means that lie within _______ of the population mean, m.
d. 100(1-x)% of all possible samples have means that lie within _______ of the population mean, m. (Hint draw a graph for the distribution of x and determine the z-scores dividing the area under the normal curve into a middle 1-x area and two outside areas of x/2.)