Question

J Morton et al examined gallbladder function before and after fundoplication - a surgery used to stop stomach contents from flowing back into the esophagus (acid reflux) - in patients with gastroesophageal reflux disease.  The authors measured gallbladder functionality by calculating the gallbladder ejection fraction (GBEF) before and after fundoplication for 12 patients.  The goal of fundoplication is to increase GBEF, which is measured as a percent.  The data are shown below.  Can we conclude that fundoplication increases GBEF functioning?  Construct a 95% confidence interval.

show work/calculations of each step

cannot use statical calculator/or programs

   Preop %     22.0    63.3      96       9.2        3.1      50.0      33        69.0        64         18.8      10           34

   Postop %    65.5    91.5      59     39.8      10.1      19.6      41        89.8        86         55.0      88         40

Answer 1

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	22	65.5	-43.5
	63.3	91.5	-28.2
	96	59	37
	9.2	39.8	-30.6
	3.1	10.1	-7
	50	19.6	30.4
	33	41	-8
	69	89.8	-20.8
	64	86	-22
	18.8	55	-36.2
	10	88	-78
	34	40	-6
Average	39.367	57.108	-17.742
St. Dev.	29.014	27.941	31.042
n	12	12	12

For the score differences, we have

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:

Ha:

This corresponds to a left-tailed test, for which a t-test for two paired samples be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=11.

Hence, it is found that the critical value for this left-tailed test is tc​=−1.796, for α=0.05 and df = 11

The rejection region for this left-tailed test is

(3) Test Statistics

The t-statistic is computed as shown in the following formula:

(4) The decision about the null hypothesis

Since it is observed that t=−1.98<tc​=−1.796, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0366, and since p=0.0366<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ1​ is less than μ2​, at the 0.05 significance level. Therefore we can conclude that fundoplication increases GBEF functioning.

Confidence interval

The 95% confidence interval is calculated as:

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