Chapter 11.1, Problem 8E

To determine

To find: Whether there is sufficient evidence to conclude that seagulls are showing preference for where they are land.

Answer to Problem 8E

Yes, there is sufficient evidence to conclude that seagulls are showing preference for where they are land.

Explanation of Solution

Given:

The proportion values are 0.56, 0.29 and 0.15.

The table is:

83

29

20

18

Formula used:

The formula to compute the test statistic is

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

Calculation:

The null and alternative hypotheses are:

  $\begin{array}{l}{H}_0:p_1=0.56,p_2=0.29,p_3=0.15\\ H_a:\text{ At least one of }{p}_i\text{ is different}\end{array}$

The calculation for test statistic is done as:

Observed value	Expected value	$\left(O-E\right)$	${\left(O-E\right)}^2$	${\left(O-E\right)}^2/E$
0.56	112	16	256	2.2857
0.29	61	3	9	0.1552
0.15	30	-19	361	12.0333
				${\displaystyle \sum {\left(O-E\right)}^2/E}=12.0333$

The test statistic is:

  $\begin{array}{c}{\chi }^2={\displaystyle \sum {\left(O-E\right)}^2/E}\\ =14.4742\end{array}$

The degree of freedom is calculated as:

  $\begin{array}{c}\text{Degree of freedom =Number of categories - 1}\\ =3-1\\ =2\end{array}$

The p-value using chi-square table at 2 degree of freedom is 0.000719

The p -value is below significance level. The null hypothesis gets rejected. there is sufficient evidence to conclude that seagulls are showing preference for where they are land.