Chapter 10.1, Problem 10.21E

To determine

To Explain: the temperature that has a significant effect on hatching and summary and conclusion.

Answer to Problem 10.21E

Degree of freedom is 2

There is not enough evidence to help the claim that there is a connection between hatching and temperature.

Explanation of Solution

Given:

	Eggs	Hatched
Cold	27	16
Neutral	56	38
Hot	104	75

Formula used:

  ${\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}$

  $E=np$

Calculation:

Assume

  $\alpha =0.05=5\%$

The null hypothesis states that there is no connection between the variables, where the alternative hypothesis states that there is a connection between the variable.

  $H_0:$ There is no connection between smoking and Drinking.

  $H_1:$ There is a connection between smoking and Drinking.

Table showing a two-way table of temperature by outcome

	Hatched	Not Hatched	Total
Cold	16	27-16=11	27
Neutral	38	59-38=18	56
Hot	75	104-75=29	104
Total	129	11+18+29=58	129+58=187

The expected frequencies $E$

  $\begin{array}{c}{E}_{11}=\frac{r_1\times c_1}{n}=\frac{27\times 129}{187}=18.6257\\ E_{12}=\frac{r_1\times c_2}{n}=\frac{27\times 58}{187}=8.3743\\ E_{21}=\frac{r_2\times c_1}{n}=\frac{56\times 129}{187}=38.6310\\ E_{22}=\frac{r_2\times c_2}{n}=\frac{56\times 58}{187}=17.3690\\ E_{31}=\frac{r_3\times c_1}{n}=\frac{104\times 129}{187}=71.7433\\ E_{32}=\frac{r_3\times c_2}{n}=\frac{104\times 58}{187}=32.2567\end{array}$

The value of the test- statistic is

  $\begin{array}{c}{\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}\\ =\frac{{\left(16-18.6257\right)}^2}{18.6257}+\frac{{\left(11-8.3743\right)}^2}{8.3743}\\ +\frac{{\left(38-38.6310\right)}^2}{38.6310}+\frac{{\left(18-17.3690\right)}^2}{17.3690}\\ +\frac{{\left(75-71.7433\right)}^2}{71.7433}+\frac{{\left(29-32.2567\right)}^2}{32.2567}\\ =1.7033\end{array}$

The degree of freedom is

  $df=(r-1)\left(c-1\right)=\left(3-1\right)\left(2-1\right)=2$

The P-Value is the probability of getting the value of the test statistic

  $P>0.25$

If the P-value is equal or less than to the significance level, then the alternative hypothesis is accepted and null hypothesis is rejected:

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

There is not enough evidence to help the claim that there is a connection between hatching and temperature.