If the average rainfall in a given area is 8 inches and the following table represents the average rainfall for four consecutive years, then the value of test statistic 't' is_____.
| Year | year 1 | year 2 | Year 3 | year 4 |
| Rainfall (inches) | 8 | 5 | 6 | 5 |
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → -2.828
This is a one-sample t-test of the sample against the population mean \( \mu=8 \), with n = 4.
\( \displaystyle \bar{x}=\frac{8+5+6+5}{4}=\frac{24}{4}=6 \)
Deviations from the mean: 2, −1, 0, −1, so
\( \displaystyle \sum(x-\bar{x})^2=4+1+0+1=6 \)
The sample standard deviation (divisor n − 1):
\( \displaystyle s=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}} \)
\( \displaystyle =\sqrt{\frac{6}{3}}=\sqrt{2} \)
\( \displaystyle t=\frac{\bar{x}-\mu}{s/\sqrt{n}} \)
\( \displaystyle =\frac{6-8}{\sqrt{2}/\sqrt{4}} \)
\( \displaystyle =\frac{-2}{\sqrt{2}/2} \)
\( \displaystyle =-\frac{4}{\sqrt{2}} \)
\( \displaystyle =-2\sqrt{2} \)
\( \displaystyle =-2.828 \)
The other common form of the statistic gives the same value. With the divisor n,
\( \displaystyle S=\sqrt{\frac{6}{4}} \)
\( \displaystyle \frac{S}{\sqrt{n-1}}=\sqrt{\frac{6}{12}}=\frac{1}{\sqrt{2}} \)
\( \displaystyle t=\frac{\bar{x}-\mu}{S/\sqrt{n-1}}=\frac{-2}{1/\sqrt{2}} \)
\( \displaystyle =-2\sqrt{2} \)