Question

There are 2 full boxes of numbered tickets. The numbers in box A have a mean = 10 and a standard deviation = 5. For box B,mean= 10,SD= 9.

A) If you draw 64 numbers (with replacement) from box A, you expect the SUM to be about _____. give or take about ______. (Show your work)

B) If you draw 100 numbers (with replacement) from box B, you expect the MEAN to be about___, give or take about ____. (Show your work.)

C) If you average 25 numbers from box A, and you average 64 numbers from box B, which do you expect to be closer to 10 ? Explain.

Answer 1

Theory -

1. In sampling with replacement the mean of all sample means equals the mean of the population:  
2. When sampling with replacement the standard deviation of all sample means equals the standard deviation of the population divided by the square root of the sample size when sampling with replacement.
3. Whatever the shape of the population distribution, the distribution of sample means is approximately normal with better approximations as the sample size, n, increases.

A)

sample mean = population mean = 10

standard deviation [mean] = Population mean / sqrt(n) = 5 / sqrt(64) = 5 / 8 = 0.625

If you draw 64 numbers (with replacement) from box A, you expect the SUM to be about 10  give or take about 0.625 .

B)

sample mean = population mean = 10

standard deviation [mean] = Population mean / sqrt(n) = 9 / sqrt(100) = 5 / 10 = 0.5

If you draw 100 numbers (with replacement) from box B, you expect the MEAN to be about 10  give or take about 0.5 .

C)

average 25 numbers from box A -

mean = 10

Sd = 5 / sqrt(25) = 5/5=1

average 64 numbers from box B -

mean = 10

SD = 9 / sqrt(64) = 9 / 8 = 1.125

we would expect numbers from Box B closes to 10,because despite it have sd greater than sd of box A, it have larger sample size.and as sample size increases our difference between our sample mean and population mean starts decreasing.

and 64 is quite enough to bring sample mean close to 10.

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