Chapter 11.2, Problem 50E

a.

To determine

To construct: The appropriate graph that shows the relationship between two variables and interpret it. Also, explain the relationship between laundry practices and the choice of detergent

Explanation of Solution

Given:

The table is:

	Soft, warm	Soft, hot	Hard, warm	Hard, hot
Standard	0.4569	0.4821	0.3818	0.4167
New	0.5431	0.5179	0.6182	0.5833

Graph:

The figure could be constructed as:

  

Interpretation:

From the above figure, it is clear that most of the people in each category are preferring the new product and it is the highest in the “Hard-Warm” group and the lowest in the “Soft-Hot: group.

b.

To determine

To check: If the sample is providing sufficient evidence to conclude that the practices of laundry and preferences of product are independent or not.

Explanation of Solution

Calculation:

The appropriate null and alternative hypotheses for the provided test could be written as:

  $\begin{array}{l}{H}_0:\text{ Two variables are independent}\\ H_a:\text{ Two variables are dependent}\end{array}$

The expected count is computed as:

	Soft, warm	Soft, hot	Hard, warm	Hard, hot
Standard	49.8079	24.0452	47.2316	30.9153
New	66.1921	31.9548	62.7684	41.0847

The test statistic is:

  $\begin{array}{c}{\chi }^2={\displaystyle \sum {\frac{\left(O-E\right)}{E}}^2}\\ =\frac{{\left(53-49.8079\right)}^2}{49.8079}+\frac{{\left(27-24.0452\right)}^2}{24.0452}+\mathrm{.......}+\frac{{\left(42-41.0847\right)}^2}{41.0847}\\ =2.058\end{array}$

The degree of freedom is calculated as:

  $\begin{array}{c}\text{Degree of freedom =}\left(\text{Number of rows - 1}\right)\left(\text{Number of column - 1}\right)\\ =\left(4-1\right)\left(2-1\right)\\ =3\end{array}$

The p-value using chi-square table at 3 degree of freedom is 0.560

The p -value is above significance level. The null hypothesis does not get rejected. Thus, there is insufficient evidence to conclude that there is an association between two variables.