Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test; i.e., describe the values of the test statistic that would result in rejection of the null hypothesis.
-A farmer has decided to use a new additive to grow his crops. He divided his farm into 10 plots and kept records of the corn yield (in bushels) before and after using the additive. The results are shown below. $\begin{array} { | l c c c c c c c c c c }
\hline \text { Plot: } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \
\hline \text { Before } & 9 & 9 & 8 & 7 & 6 & 8 & 5 & 9 & 10 & 11 \
\text { After } & 10 & 9 & 9 & 8 & 7 & 10 & 6 & 10 & 10 & 12 \
\hline
\end{array}$
You wish to test the following hypothesis at the 10 percent level of significance. $\mathrm { H } _ { 0 } : \mu _ { \mathrm { d } } = 0 \text { against } \mathrm { H } _ { 1 } : \mu _ { \mathrm { d } } \neq 0$
What decision rule would you use?
A) Reject $\mathrm { H } _ { 0 }$ if test statistic is greater than $- 1.833$ or less than $1.833$.
B) Reject $\mathrm { H } _ { 0 }$ if test statistic is less than $- 1.833$.
C) Reject $\mathrm { H } _ { 0 }$ if test statistic is greater than $1.833$.
D) Reject $\mathrm { H } _ { 0 }$ if test statistic is less than $- 1.833$ or greater than $1.833$.

Question

Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test; i.e., describe the values of the test statistic that would result in rejection of the null hypothesis.
-A farmer has decided to use a new additive to grow his crops. He divided his farm into 10 plots and kept records of the corn yield (in bushels) before and after using the additive. The results are shown below. $\begin{array} { | l c c c c c c c c c c } 
\hline \text { Plot: } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \
\hline \text { Before } & 9 & 9 & 8 & 7 & 6 & 8 & 5 & 9 & 10 & 11 \
\text { After } & 10 & 9 & 9 & 8 & 7 & 10 & 6 & 10 & 10 & 12 \
\hline
\end{array}$
You wish to test the following hypothesis at the 10 percent level of significance. $\mathrm { H } _ { 0 } : \mu _ { \mathrm { d } } = 0 \text { against } \mathrm { H } _ { 1 } : \mu _ { \mathrm { d } } \neq 0$
 What decision rule would you use? 
A) Reject $\mathrm { H } _ { 0 }$ if test statistic is greater than $- 1.833$ or less than $1.833$.
B) Reject $\mathrm { H } _ { 0 }$ if test statistic is less than $- 1.833$.
C) Reject $\mathrm { H } _ { 0 }$ if test statistic is greater than $1.833$.
D) Reject $\mathrm { H } _ { 0 }$ if test statistic is less than $- 1.833$ or greater than $1.833$.

Quizplus · Accepted Answer

We are given a paired sample with n = 10 observations. Since we want to test the null hypothesis that the mean difference in yield before and after using the additive is zero, we need to calculate the differences between each paired observation (i.e. after-before) and find the mean and standard deviation of these differences.

The test statistic for this paired samples t-test is:

$$t = \frac{\bar{d} - \mu_{d}}{s/\sqrt{n}}$$

where $\bar{d}$ is the mean of the differences, $\mu_{d}$ is the hypothesized population mean difference (which is zero for the null hypothesis), $s$ is the standard deviation of the differences, and $n$ is the sample size.

Using the data provided in the table, we find that the mean difference is $\bar{d} = 0.9$, and the standard deviation of the differences is $s=1.63299$.

Substituting these values into the formula for the test statistic, we get:

$$t = \frac{0.9 - 0}{1.63299/\sqrt{10}} = 1.8498$$

Since we are conducting a two-tailed test at the 10% level of significance, we need to find the critical values of t that correspond to a 5% probability in each tail of the distribution. Using a t-table or a calculator function, we find that the critical values are $\pm 1.833$ for 9 degrees of freedom (df = n-1).

Since our calculated t-value of 1.8498 falls outside the range of -1.833 to 1.833, we reject the null hypothesis and conclude that there is evidence to suggest that the additive has a significant effect on corn yield.

Therefore, the decision rule is to reject $\mathrm{H}_0$ if the test statistic is less than -1.833 or greater than 1.833, which corresponds to choice D.

Determine the Decision Criterion for Rejecting the Null Hypothesis in the Given