Question

Find the probability that the sample mean is less than 984 if a sample of size 100 is taken from a population with a mean of 1000 and a standard deviation of 300.

Answer 1

Given Data

Sample Size = n =100

Population mean = μ = 1000

Sample mean = x = 984

Sample standard deviation = s = 300

To calculate what is probabilty that sample mean will be less than 984, we have to find z score for given distribution which we assume is normal as sample size is large. z score tells us how much standard deviation we are away from mean. so for given value we calculate z score first and then using excel or z table for right tailed z test (as we have to find probabilty for value falling less than 984) we will find p value.

For z score

Calculation of p value from z score using excel fromula =NORM.S.DIST(-0.5333,TRUE)

p= 0.2969

So probabilty that sample mean is less than 984 is 29.69%

you can verify from z table for rleft tailed test given below where I have shown corresponding values