finding number of standard deviations between sample and population mean

Jul 2010
160
8
Suppose that the mean mileage of all pickup trucks of a particular make is unknown and will be estimated from a random sample of 36 trucks. The sample mean mileage was found to be 6 litres per 100 km. Assume that the standard deviation of the mileages of all trucks is .5 litres per 100 km. There is a 0.95 probability that the mean of the sample will be within z SDs of the unknown mileage. What is z?

I believe I find the z-score associated with 0.95 but I'm not sure if I need to do something with the sample mean or SD.
 
Dec 2009
3,120
1,342
Suppose that the mean mileage of all pickup trucks of a particular make is unknown and will be estimated from a random sample of 36 trucks. The sample mean mileage was found to be 6 litres per 100 km. Assume that the standard deviation of the mileages of all trucks is .5 litres per 100 km. There is a 0.95 probability that the mean of the sample will be within z SDs of the unknown mileage. What is z?

I believe I find the z-score associated with 0.95 but I'm not sure if I need to do something with the sample mean or SD.
A 0.95 probability that the sample mean is within z standard deviations
means that you need to locate the two 2.5% regions at the tails.

Hence, you need the z-score corresponding to 0.975 instead.
Also, as you have the "population" SD

\(\displaystyle Z=\displaystyle\huge\frac{sample\ mean-\mu}{\frac{\sigma}{\sqrt{36}}}\)

where \(\displaystyle \mu\) is the population mean.

The above calculation is taken for sample mean greater than population mean.
The graph is symmetrical however.
 
Jul 2010
160
8
The area under the normal distribution curve for 0.25% is z=-1.96 and for 97.25% is z=1.96. The answer to my question is 1.96. So why do I need \(\displaystyle Z=\displaystyle\huge\frac{sample\ mean-\mu}{\frac{\sigma}{\sqrt{36}}}\)?
 
Dec 2009
3,120
1,342
The area under the normal distribution curve for 0.25% is z=-1.96 and for 97.25% is z=1.96. The answer to my question is 1.96. So why do I need \(\displaystyle Z=\displaystyle\huge\frac{sample\ mean-\mu}{\frac{\sigma}{\sqrt{36}}}\)?
You don't need it for Z.
You could use that Z to get a reading for population mean if needed.