In: Statistics and Probability
Suppose a random sample of n = 16 observations is selected from a population that is normally distributed with mean equal to 105 and standard deviation equal to 11. (use SALT)
(a) Give the mean and the standard deviation of the sampling distribution of the sample mean
x.
mean | |
standard deviation |
(b) Find the probability that x exceeds 109. (Round your
answer to four decimal places.)
(c) Find the probability that the sample mean deviates from the
population mean μ = 105 by no more than 5. (Round your
answer to four decimal places.)
Solution :
Given that ,
mean = = 105
standard deviation = = 11
n = 16
a)
_{} = = 105 and
_{} = / n = 16 / 16 = 4
b)
P( > 109) = 1 - P( < 109)
= 1 - P(( - _{} ) / _{} < (109 - 105) / 5)
= 1 - P(z < 1)
= 1 - 0.8413 Using standard normal table.
= 0.1587
Probability = 0.1587
c)
P(100 < < 110) = P((100 - 105) / 4<( - _{}) / _{} < (110 - 105) / 4))
= P(-1.25 < Z < 1.25)
= P(Z < 1.25) - P(Z < -1.25) Using standard normal table,
= 0.8944 - 0.1056
= 0.7888
Probability = 0.7888