Chapter 11, Problem 3CRE

(a)

To determine

To construct: The two way table of cardiac events.

Explanation of Solution

Calculation:

Given information:

Total number of people $=107$

Number of people in stress management program $=33$

Number of people suffered cardiac events in stress management program $=3$

Number of people in the exercise group $=34$

Number of people suffered cardiac events in the exercise group $=7$

Number of people in usual care $=40$

Number of people suffered cardiac events in usual care $=12$

Graph:

The two way table will be

	Stress management	Exercise group	Usual care	Total
Suffered cardiac events	$3$	$7$	$12$	$22$
Did not suffered cardiac events	$30$	$27$	$28$	$85$
Total	$33$	$34$	$40$	$107$

(b)

To determine

To find: The success rates of the three treatments.

Answer to Problem 3CRE

Stress management $=0.9091$

Exercise program $=0.7941$

Usual care $=0.70$

Explanation of Solution

Formula used:

  $\text{Success rate = }\frac{\text{Number of people who did not suffered cardiac events in the group}}{\text{total number of people in the group}}$

Calculation:

Success rates

For stress management:

  $=\frac{30}{33}=0.9091$

For exercise program:

  $=\frac{27}{34}=0.7941$

For usual care:

  $=\frac{28}{40}=0.7$

(c)

To determine

To check: Whether there is any significant difference among the success rates for the three treatments or not.

Answer to Problem 3CRE

There is no significant difference among the success rates for the three treatments.

Explanation of Solution

Formula used:

The chi square statistic is

  ${\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}~{\chi }_{\left(r-1\right)\times \left(c-1\right)}}$

Calculation:

The expected count of the number of people can be calculated as follows

	Stress management	Exercise group	Usual care	Total
Suffered cardiac events	$\frac{22\times 33}{107}=6.79$	$\frac{22\times 34}{107}=6.99$	$\frac{22\times 40}{107}=8.22$	$22$
Did not suffered cardiac events	$\frac{85\times 33}{107}=26.21$	$\frac{85\times 34}{107}=27.01$	$\frac{85\times 40}{107}=31.78$	$85$
Total	$33$	$34$	$40$	$107$

The null and alternate hypotheses are

  $H_0:$ There is no difference in the actual success rates for the three treatments.

  $H_1:$ There is a difference in the actual success rates for the three treatments.

Test statistic

Under null hypothesis

  $\frac{{\left(3-6.79\right)}^2}{6.79}+\frac{{\left(7-6.99\right)}^2}{6.99}+\frac{{\left(12-8.22\right)}^2}{8.22}+\frac{{\left(30-26.21\right)}^2}{26.21}+\frac{{\left(27-27.01\right)}^2}{27.01}+\frac{{\left(28-31.78\right)}^2}{31.78}~{\chi }_2$

The test value is $4.8514$

P value:

  $P\left({\chi }_2>4.8514\right)=0.0884$

Conclusion:

As p value is greater than $0.05$ (level of significance), so there is insufficient evidence to reject null hypothesis at $5\%$ level of significance and thus conclude that there is no difference in the actual success rates for the treatments.