Chapter 11.1, Problem 15E

To determine

To find: Whether there is sufficient evidence to conclude that the data is consistent with the belief

Answer to Problem 15E

No, there is insufficient evidence to conclude that the data is consistent with the belief

Explanation of Solution

Given:

The data set is:

Count
321
360
367
374
383
402
392
329
331
354
376
355

Formula used:

The formula to compute the test statistic is

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

Calculation:

The null and alternative hypotheses are:

  $\begin{array}{l}{H}_0:p_1=p_2=p_3=\mathrm{........}=p_{12}=0.0833\\ H_a:\text{ At least one of }{p}_i\text{ is incorrect}\end{array}$

The calculation for test statistic is done as:

Observed value	Expected value	$\left(O-E\right)$	${\left(O-E\right)}^2$	${\left(O-E\right)}^2/E$
321	362	-41	1681	4.46436
360	362	-2	4	0.011
367	362	6	25	0.0691
374	362	12	144	0.3978
383	362	21	441	1.2182
402	362	40	1600	4.4199
392	362	30	900	2.4862
329	362	-22	1089	3.0083
331	362	-31	961	2.6547
354	362	-8	64	0.1768
376	362	14	196	0.5414
355	362	-7	49	0.1354
				${\displaystyle \sum {\frac{(}{}}^{}}$ O−E)E2=19.7624

The test statistic is:

  $\begin{array}{c}{\chi }^2={\displaystyle \sum {\frac{\left(O-E\right)}{E}}^2}\\ =19.7624\end{array}$

The degree of freedom is calculated as:

  $\begin{array}{c}\text{Degree of freedom =Number of categories - 1}\\ =12-1\\ =11\end{array}$

The p-value using chi-square table at 11 degree of freedom is 0.00054

The p -value is below significance level. The null hypothesis gets rejected. Thus, there is insufficient evidence to conclude that the data is consistent with the belief