Question

Examine the required sample size needed to be able to

estimatethe mean dollars that each consumer will spend each month.
Desireis to be within plus or minus $10 of the true mean with a
98%confidence level. The standard deviation is thought to
be500.
What are the trade-offs that will occur when you lower
thesample level to 2000?

Answer 1

`n=((2.33(500))/(10))^2 =13572.25 \approx 13573`

And if we use a sample level of n =2000 the margin of error would be higher as we can see here:

`ME=2.33(500)/(√(2000))=26.05`

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=500` represent the population standard deviation assumed

n represent the sample size (variable of interest)

Confidence =98% or 0.98

ME = 10 represent the margin of error desired

The margin of error is given by this formula:

`ME=z_(\alpha/2)(\sigma)/(√(n))` (a)

And on this case we have that ME =10 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

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

The critical value for 98% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.01;0;1)", and we got
`z_(\alpha/2)=2.33`, replacing into formula (b) we got:

`n=((2.33(500))/(10))^2 =13572.25 \approx 13573`

So the answer for this case would be n=13573 rounded up to the nearest integer

And if we use a sample level of n =2000 the margin of error would be higher as we can see here:

`ME=2.33(500)/(√(2000))=26.05`