An analyst from an energy research institute in California wishes to estimate the 90% confidence interval for the average price of unleaded gasoline in the state. In particular, she does not want the sample mean to deviate from the population mean by more than $0.05. What is the minimum number of gas stations that she should include in her sample if she uses the standard deviation estimate of $0.31, as reported in the popular press

So the answer for this case would be n=104 rounded up to the nearest integer

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

represent the sample mean for the sample  

population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Solution to the problem

The margin of error is given by this formula:

   (a)

And on this case we have that ME =0.31 and we are interested in order to find the value of n, if we solve n from equation (b) we got:

  (b)

The critical value for 90% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.05;0;1)", and we got , replacing into formula (b) we got:

So the answer for this case would be n=104 rounded up to the nearest integer