Question

Assume that the one-way commute time of an UoU student from his house to school is a normally distributed random variable which we will call X. Furthermore, assume that the population standard deviation of X is σ = 10 minutes. Let μ be the unknown population mean for X:

Experimental design: Determine a minimum sample size such that we will be 95 % confident that the error will not exceed 5 minutes when the sample average x is used to estimate μ. Let n denote this sample size

Answer 1

`n=((1.960(10))/(5))^2 =15.36 \approx 16`

So the answer for this case would be n=16 rounded up to the nearest integer

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".

`\bar X` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=10` represent the sample standard deviation

n represent the sample size

ME=5 represent the margin of error

Solution to the problem

The confidence interval for the mean is given by the following formula:

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

The margin of error is given by this formula:

`ME=z_(\alpha/2)(\sigma)/(√(n))` (2)

And on this case we have that ME =5 and we are interested in order to find the value of n, if we solve n from equation (2) we got:

`n=((z_(\alpha/2) \sigma)/(ME))^2` (3)

The critical value for 95% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.025;0;1)", and we got
`z_(\alpha/2)=1.96`, replacing into formula (3) we got:

`n=((1.960(10))/(5))^2 =15.36 \approx 16`

So the answer for this case would be n=16 rounded up to the nearest integer