Determine Sample Size

Sample size

We determine the sample size in to determine the mean value to be spent by consumers each month given that the sample should yield a plus or minus 10 from the mean with a 98 confidence interval and we are given that the standard deviation is equal to 500 , We construct a confidence interval as follows ? x (x (sd . Z )] y Where x ' is the mean , sd is the standard deviation , Z is the value from the T table determined by confidence level selected and Y is the confidence

level , in our case therefore we will formulate our confidence interval as follows

[ (x '- (500 . Z ? x (x (500 . Z )] 98

We are also provided with information that the sample should yield a mean where at 98 there should be a plus or minus 10 , for this reason the x ' from our above function will deviate plus or minus ten , therefore (500 . Z 10 such that the confidence interval will be as follow

[ (x '- (10 ? x (x (10 )] 98

We now solve (500 . Z 10

where Z 10 /500

Z 0 .02

From the table a sample of 10000 at 98 level of test yields 0 .025 which approximately yields a plus and minus ten of the confidence interval , if we choose the 10 , 000 sample size then our confidence interval at 98 will be as follows

[ (x '- (500 . 0 .02507 ? x (x (500 . 0 .02507 )] 98 ] 98

The company is planning to spend 10 ,000 on the sample however it costs 5 for each sample and therefore there is a need to reduce the sample size , using the 10 ,000 sample size trhe cost would be 10 ,000 X 5 50 ,000 , in to spend 10 ,000 there is a need to choose a sample size equal to 10 ,000 /5 2 ,000

Choosing a sample size of 2 ,000 will result into a larger confidence interval , a sample of 2 ,000 at 98 confidence interval will yield the following confidence interval

[ (x '- (12 .53602 ? x (x (12 .53603 )] 98

From the calculations in the first instance where the sample size was 10 ,000 the data deviates 12 .53477 from the mean at 98 confidence interval , in the second case where the sample size is 2 ,000 the data deviates 12 .53602 from the mean at 98 confidence interval , for this reason therefore when we use a smaller sample size we have a larger confidence interval

References

Allan Bluman (2002 ) Elementary Statistics : A Step by Step Approach McGraw Hill press , New York

98 (x '- (sd . Z ) Mean (0 (x (sd . Z...