Consider a large population of interest. It's distribution is normal and it's mean is 176 and standard deviation is 101. Let X = a single observation.
(Round all probabilities to four decimals)
a) Find P(X < 181):
A random sample of 148 is taken. Let X¯¯¯X¯ = sample average of the observations.
b) Find P(X¯¯¯X¯ < 181):
Consider another large population of interest. It's distribution is unknown and it's mean is 176 and it standard deviation is 101. Let Y = a single observation.
A random sample of 148 is taken. Let Y¯¯¯Y¯= sample average of the observations.
c) Find P(Y¯¯¯Y¯ < 181):
(a)
= 176
=101
To find P(X<181):
Z = (181 - 176)/101 = 0.0495
Table of Area Under Standard Normal Curve gives area = 0.0199
So,
P(X<181) = 0.5 + 0.0199 = 0.5199
(b)
= 176
=101
n = 148
SE = /
= 101/ = 8.3021
To find P(<181):
Z = (181 - 176)/8.3021 = 0.6023
Table of Area Under Standard Normal Curve gives area = 0.2257
So,
P(<181) = 0.5 + 0.2257 = 0.7257
(c)
By Central Limit Theorem, the sampling distribution of sample means is normal distribution even if the population distribution is not normal distribution.
So,
= 176
=101
n = 148
SE = /
= 101/ = 8.3021
To find P(<181):
Z = (181 - 176)/8.3021 = 0.6023
Table of Area Under Standard Normal Curve gives area = 0.2257
So,
P(<181) = 0.5 + 0.2257 = 0.7257
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