Chapter 24, Problem 38E

To determine

To investigate the question of whether the mean length of cuckoo eggs is the same for different species and state your conclusion.

Answer to Problem 38E

The mean length of cuckoo eggs is the same for different species.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=23.0667\\ {\overline{x}}_2=22.575\\ n_1=12\\ n_2=16\\ s_1=1.1392\\ s_2=0.6846\end{array}$

Now, let us define the hypotheses for testing:

  $\begin{array}{l}{H}_0:{\mu }_1={\mu }_2\\ H_1:{\mu }_1\ne {\mu }_2\end{array}$

Thus, the value of test statistics is:

  $\begin{array}{l}t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ =\frac{23.0667-22.575}{\sqrt{\frac{{1.1392}^2}{12}+\frac{{0.6846}^2}{16}}}\\ =1.33\end{array}$

And the value of degree of freedom is:

  $\begin{array}{l}df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})}{\frac{{(\raisebox{1ex}{$s_1^2$}\!\left/ \!\raisebox{-1ex}{$n_1$}\right.)}^2}{n_1-1}+\frac{{(\raisebox{1ex}{$s_2^2$}\!\left/ \!\raisebox{-1ex}{$n_2$}\right.)}^2}{n_2-1}}\\ =16\end{array}$

Thus, the P-value will be:

  $P>0.20$

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

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

Thus, there is not sufficient evidence to support the claim of a difference. So, the mean length of cuckoo eggs is the same for different species.