Chapter 10.3, Problem 86E

To determine

To explain do these data provide convincing evidence at the $\alpha =0.05$ level that drying barely seeds in a kiln increases the yield of barley on average.

Answer to Problem 86E

There is no convincing evidence at the $\alpha =0.05$ level that drying barely seeds in a kiln increases the yield of barley on average.

Explanation of Solution

It is given that:

  $\begin{array}{l}\alpha =0.05\\ n=11\end{array}$

Let us find the difference between the inside time and the drive-thru time for each visit as:

Plot	Regular	Kiln	Difference
1	1903	2009	-106
2	1935	1915	20
3	1910	2011	-101
4	2496	2463	33
5	2108	2180	-72
6	1961	1925	36
7	2060	2122	-62
8	1444	1482	-38
9	1612	1542	70
10	1316	1443	-127
11	1511	1535	-24

The mean value of the difference is as:

  $\begin{array}{l}\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\\ =\frac{106+20-101+33-72+36-62-38+70-127-24}{11}\\ =\frac{-371}{11}\\ =-33.7273\end{array}$

The standard deviation of the difference is as follows:

  $\begin{array}{l}s=\sqrt{\frac{{\displaystyle \sum {(x-\overline{x})}^2}}{n-1}}\\ =\sqrt{\frac{\begin{array}{}$ {(106-(-33.7273))}^2+{(20-(-33.7273))}^2+{(-101-(-33.7273))}^2+{(33-(-33.7273))}^2+$$ \mathrm{...}+{(-24-(-33.7273))}^2$\end{array}}{11-1}}\\ =66.1711\end{array}$

Given claim is that: increases for kiln.

The claim is either null hypothesis or an alternative hypothesis.

  $\begin{array}{l}{H}_0:{\mu }_D=0\\ H_a:{\mu }_D<0\end{array}$

The test statistics value will be:

  $\begin{array}{l}t=\frac{{\overline{x}}_D-{\mu }_D}{\frac{s_D}{\sqrt{n}}}\\ =\frac{-33.7273-0}{\frac{66.1711}{\sqrt{11}}}\\ =-1.690\end{array}$

And the degree of freedom will be:

  $\begin{array}{l}df=n-1\\ =11-1\\ =10\end{array}$

As we know that the P -value is the probability of obtaining the sample results or results more extreme when the null hypothesis is true.

  $0.05<P<0.10$

And we know that if the P -value is less than or equal to the significance level then the null hypothesis is rejected, then,

  $P>0.05\Rightarrow \text{Fail to Reject }{H}_0$

Thus, we conclude that there is no convincing evidence at the $\alpha =0.05$ level that drying barely seeds in a kiln increases the yield of barley on average.