conclusion using p-value and critical value approaches

Question

A sample of 9 days over the past six months showed that a clinic treated the following numbers of patients: 24, 26, 21, 17, 16, 23, 27, 18, and 25. If the number of patients seen per day is normally distributed, would an analysis of these sample data provide evidence that the variance in the number of patients seen per day is less than 10? Use α = .025 level of significance. What is your conclusion using p-value and critical value approaches. Is the conclusion different in both the cases?

Answer 1

 

Hypothesis Formation

H₀: σ =10

H₁: σ < 10

Test Statistics

χ² = (n-1).S²/ σ²

Critical Region

Reject H0 in favor of alternative if χ² test statistic lesser than the critical value of χ²

i.e χ²test statistic < critical χ²

Critical value of χ² at 0.025 Significance Level for single tail test

Df = n – 1 = 9 – 1 = 8

Critical value of χ² with df 8 and alpha 0.025 = 2.18

Computation

Data (X)	X – X-bar	(X-X-bar)2
24	2.111111	4.45679
26	4.111111	16.90123
21	-0.88889	0.790123
17	-4.88889	23.90123
16	-5.88889	34.67901
23	1.111111	1.234568
27	5.111111	26.12346
18	-3.88889	15.12346
25	3.111111	9.679012

 

Sum of (X-X-bar)² = 132.89

S² = 132.89/9-1

     = 16.61 

χ² = (9-1)*16.61/10

    = 13.29

Decision

As χ² statistic is not less than critical value, therefore we can’t say that variance is less than 10. P-value for critical value is 0.01 and it is approximately found from χ² table.  P-value is greater than our tolerance for ambiguity therefore we can’t that variance is significantly lower than 10.