Question 1081974: A random sample of the number of farms (in thousands) in various states follows. Estimate the mean number of farms per state with 99% confidence. Assume o=31. Round intermediate and final answers to one decimal place. Assume the population is normally distributed.
47 50 40 109 78 6 52 21 15 7 68
16 48 79 44
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! With 99% confidence, the z critical value is approximately 2.576
Use a calculator or a table to determine this. If you decide to use the table that I've linked, look at the "99%" in the bottom row. Then look at the value 2.576 just above it.
So we know that z = 2.576
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Now let's compute the sample mean (xbar) of the given data set. Simply add up the values and divide by 15. We divide by 15 because there are 15 values in the data set
Add up the values: 47+50+40+109+78+6+52+21+15+7+68+16+48+79+44 = 680
Divide the sum by 15
680/15 = 45.333333
Approximate to 6 decimal places
So xbar = 45.333333 roughly
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We'll use these formulas
L = xbar - z*sigma/sqrt(n)
U = xbar + z*sigma/sqrt(n)
Where
xbar = sample mean
z = critical value
sigma = population standard deviation
n = sample size
L = lower bound of confidence interval
U = upper bound of confidence interval
In this case,
xbar = 45.333333 (approximate; see above)
z = 2.576 (approximate; see above)
sigma = 31 (given in the instructions)
n = 15 (since there are fifteen values given)
L = unknown value
U = unknown value
The values L and U are unknown for now, but we can use the other values to find L and U.
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Let's compute the lower boundary L
L = xbar - z*sigma/sqrt(n)
L = 45.333333 - 2.576*31/sqrt(15)
L = 24.714602
L = 24.7
Let's compute the upper boundary U
U = xbar + z*sigma/sqrt(n)
U = 45.333333 + 2.576*31/sqrt(15)
U = 65.9520635396493
U = 66.0
After rounding to 1 decimal place, the 99% confidence interval is (L, U) = (24.7, 66.0)
We're 99% confident that the mean (mu) is between 24.7 and 66.0
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Answer: (24.7,66.0) which is a confidence interval (not an ordered pair).
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