Suppose that left arm lengths in cm of the Dalhousie student

Suppose that left arm lengths (in cm) of the Dalhousie student population is found to be normally distributed with a mean of 70 cm. and standard deviation of 8 cm. If this population is subjected to repeated random sampling with replacement, and with a constant sample size of 10:

a.) What is the expected value of the sample mean (in cm)?
(Give answer to nearest whole number.)

b.) What will be the variance of the sample mean ?
(Give decimal answer to one place past the decimal.)

c.) What is the probability that the mean of a randomly selected sample is less
than or equal to 65 cm. (Give decimal answer to three places past the decimal, with no leading zero.)

Solution

a.) What is the expected value of the sample mean (in cm)?
(Give answer to nearest whole number.)

sample mean=70 (stay the same)

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b.) What will be the variance of the sample mean ?
(Give decimal answer to one place past the decimal.)

variance= s^2/vn

=8^2/10 =6.4

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c.) What is the probability that the mean of a randomly selected sample is less
than or equal to 65 cm. (Give decimal answer to three places past the decimal, with no leading zero.)

P(xbar<65) = P((xbar-mean)/(s/vn) <(65-70)/6.4)

=P(Z<-0.78) = 0.218 (from standard normal table)