Suppose we are testing the consistency of the rate of fluid flow (flow-rate) in
ID: 2924890 • Letter: S
Question
Suppose we are testing the consistency of the rate of fluid flow (flow-rate) in gallons per hou r(gal/hr) of shower heads for an upcoming shipment. The population under consideration is theN= 100shower heads which comprise the shipment. The distribution which the flow-rates follow is unknown, but wedo know (from previous testing) that the population mean is120 gal/hr(at 80PSI, or pounds per squareinch) and the population variance is 25gal2/hr2.
1. State the approximate distribution of the sample mean if it is based on a simple random sample with replacement of size n= 80, or explain why it is impossible to approximate the distribution.
2. State the approximate distribution of the sample mean if it is based on a simple random sample without replacement of size n= 80, or explain why it is impossible to approximate the distribution.
3. State the approximate distribution of the sample mean if it is based on a simple random sample with replacement of size n=2, or explain why it is impossible to approximate the distribution.
Explanation / Answer
1. State the approximate distribution of the sample mean if it is based on a simple random sample with replacement of size n= 80, or explain why it is impossible to approximate the distribution.
Solution:
We are given,
Population size = N = 100
Sample size = n = 80
Population mean = µ = 120
Population Variance = ^2 = 25
Population SD = = sqrt(25) = 5
When sample is taken by using simple random sample with replacement, then unbiased estimator of the population mean is sample mean and unbiased estimator for variance is given as below:
Estimator of population variance = s^2/n
Estimator of population SD = s/sqrt(n)
The approximate distribution of the sample mean is approximate normal distribution with µ and ,
Where, µ = 120 and = 5/sqrt(80) = 0.559017
Xbar è N(120, 0.5590)
2. State the approximate distribution of the sample mean if it is based on a simple random sample without replacement of size n= 80, or explain why it is impossible to approximate the distribution.
Solution:
We are given,
Population size = N = 100
Sample size = n = 80
Population mean = µ = 120
Population Variance = ^2 = 25
Population SD = = sqrt(25) = 5
When sample is taken by using simple random sample with replacement, then unbiased estimator of the population mean is sample mean and unbiased estimator for variance is given as below:
Estimator of population variance =[(N – n)/N]* s^2/n = [(100 – 80)/100]*25/80 = 0.0625
Estimator of population SD = sqrt(0.0625) = 0.25
Xbar è N(120, 0.25)
3. State the approximate distribution of the sample mean if it is based on a simple random sample with replacement of size n=2, or explain why it is impossible to approximate the distribution.
Solution:
Here, we are given a very small sample size as n = 2 and in the case of small or very small sample size, the distribution of the sample mean does not follows an approximate normal distribution or t distribution. The best fit of the normal distribution is achieved only when sample size is adequate and large enough. For larger sample size, there would be greater reliability of using normal distribution.