Question 1141455: We want to estimate the population mean within 18, with a 90% level of confidence. The population standard deviation is estimated to be 51. 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 jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Answer: 22
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Work Shown:
At 90% confidence, the z critical value is roughly z = 1.645
To find this value, I used this table.
More specifically, I looked at the row starting with Z (near the very bottom of the table) and looked above the 90% confidence level to find the value 1.645
Use of the invNorm function on your TI83 or TI84 calculator will yield the same approximate results (though more accurately of course).
The standard deviation given is sigma = 51
The error desired is E = 18
The values we will use are
z = 1.645 (approximate)
sigma = 51
E = 18
and they will be plugged into the formula below (see this site for more details). For each computation, I am rounding the values to 2 decimal places.
n = (z*sigma/E)^2
n = (1.645*51/18)^2
n = (83.895/18)^2
n = (83.90/18)^2
n = (4.66111111111111)^2
n = (4.66)^2
n = 21.7156 .... see note below
n = 22
note: inputting (1.645*51/18)^2 directly into a scientific calculator yields the approximate result 21.7233673611111, which is fairly close to what we get above (21.7156). The result 21.7233673611111 is a lot more accurate to the true value of n. In the end however, we'll be rounding to a whole number, so the finer decimal details don't matter in this case.
Further information: When rounding to the nearest whole number, you will always round up instead of down. This is true for something like 21.7 or 21.00007; it doesn't matter how close the decimal value is to 21, we will always bump up to 22. This is to ensure we clear the hurdle needed. If we go with n = 21, then the sample size is too small because it leaves out the decimal portion. Going with n = 22 ensures that we overshoot our goal. In the second link I posted, the third example shows n = 51.1 leading to n = 52, which is further evidence of rounding up even though 51.1 is much closer to 51 than 52.
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