SOLUTION: We want to estimate the population mean within 8, with a 80% level of confidence. The population standard deviation is estimated to be 21. How large a sample is required? (Round th

Question 1118451: We want to estimate the population mean within 8, with a 80% level of confidence. The population standard deviation is estimated to be 21. How large a sample is required? (Round the intermediate calculation to 2 decimal places. Round the final answer to the nearest whole number.)
Sample size :

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
at 80% confidence level, the critical alpha will be plus or minus .2/2 = .1.

at a critical alpha of .1, the critical z-score will be plus or minus 1.281551567 which you are instructed to round to two decimal places, making it plus or minus 1.28.

the standard error is equal to the standard deviation divided by sample size.

that would make the standard error equal to 21 / sqrt(n), where n is the sample size.

the formula for z-score is z = (x-m) / s

in this case:

z = the critical z-score
x = the raw score
m equal the mean
s equal the standard error.

on the high side, the formula becomes 1.28 = (x-m) / (21/sqrt(n)).

since (x-m) on the high side equals 8,the formula becomes 1.28 = 8 / (21/sqrt(n)).

multiply both sides of this equation by (21/sqrt(n)) and you get 1.28 * (21/sqrt(n)) = 8.

multiply both sides of this equation by sqrt(n) and divide both sides of this equation by 8 to get 1.28 * 21 / 8 = sqrt(n).

solve for n to get n = 11.2896.

tha's the sample size required.

round it to the nearest whole integer to get 11.

that's your solution.

to test this solution, it is recommended to use the unrounded values rather than the rounded values.

this way you'll get a range of plus or minus 8 exactly, or extremely close, if not.

the critical z-score is actually 1.281551567.

the sample size is actually 11.31698622.

the standard error is 21 / sqrt(11.31698622) = 6.242433164.

the formula is z = (x-m) / s.

z = plus or minus 1.281551567
s = 6.242433164
m can be any randomly generated number, such as 3524.

when z = plus, the formula becomes 1.281551567 = (x-3524) / 6.242433164.

solve for x to get x = 1.281551567 * 6.242433164 + 3524 = 3532

when z = minus, the formula becomes -1.281551567 = (x-3524) / 6.242433164.

solve for x to get x = -1.281551567 * 6.242433164 + 3524 = 3516.

3532 - 3524 = 8
3516 - 3524 = -8

the difference is plus or minus 8.

it will always be plus or minus 8 regardless of the mean because 1.281551567 * 6.242433164 = 8.

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