Chapter 11, Problem 13PT

a.

To determine

To find: Whether there is sufficient evidence to conclude that there exists an association between education level and status of smoking.

Answer to Problem 13PT

Yes, there is enough evidence to conclude that there exists an association between education level and status of smoking.

Explanation of Solution

Given:

The table is:

Education	Non smoker	Former	Moderate	Heavy
Primary School	56	54	41	36
Secondary School	37	43	27	32
University	53	28	36	16

Calculation:

The hypotheses are:

  $\begin{array}{l}{H}_0:\text{No relationship exists between two variables}\text{.}\\ H_a:\text{ Relationship exists between two variables}\text{.}\end{array}$

The calculation can be done as:

		Non smoker	Former	Moderate	Heavy	Total
Primary School	Observed	56	54	41	36	187
	Expected	59.48	50.93	42.37	34.22	187.00
Secondary School	Observed	37	43	27	32	139
	Expected	44.21	37.85	31.49	25.44	139.00
University	Observed	53	28	36	16	133
	Expected	42.31	36.22	30.14	24.34	133.00
Total	Observed	146	125	104	84	459
	Expected	146.00	125.00	104.00	84.00	459.00

The test statistic is:

  $\begin{array}{c}{\chi }^2=\frac{{\left(59-59.48\right)}^2}{59.48}+\frac{{\left(54-50.93\right)}^2}{50.93}+\mathrm{......}+\frac{{\left(36-34.22\right)}^2}{34.22}\\ =13.30\end{array}$

The degree of freedom is calculated as:

  $\begin{array}{c}\text{Degree of freedom =}\left(\text{Number of rows-1}\right)\left(\text{Number of columns-1}\right)\\ =\left(3-1\right)\left(4-1\right)\\ =6\end{array}$

The p-value using chi-square table at 6 degree of freedom is 0.384.

The p -value is above significance level. The null hypothesis does not get rejected. There is enough evidence to be in favor of claim at 5% significance level.

b.

To determine

To find: The cell that is mostly contributing to the relationship.

Explanation of Solution

Calculation:

The expected count for the cell “Heavy University” is calculated as:

  $\begin{array}{c}\text{Contribution = }\frac{{\left(16-24.3399\right)}^2}{24.3399}\\ =2.858\end{array}$

Thus, the highest contribution is of “Heavy University”.