Question

Kate has been asked to do a statistical analysis of the results of trials on two drugs, Treatment 1 relieved the symptoms of a complaint in 47 out of the 75 patients on whom it was tried, while treatment 2 relieved the symptoms of the same complaint in 43 out of 62 patients.

Kate has been asked to test H₀: p₁ = p₂ vs H₁: p₁ > p₂, where p₁ and p₂ are the true proportions of patients who experience relief of symptoms from treatments 1 and 2 respectively. Kate should report that

		p1 is significantly greater than p2 at the 2.5% level of significance, but not at the 1% level
		p1 is significantly greater than p2 at the 5% level of significance, but not at the 2.5% level
		p1 is significantly greater than p2 at the 1% level of significance
		p1 is significantly greater than p2 at the 10% level of significance, but not at the 5% level
		there is not a significant difference between the true proportions at the 10% level of significance

Answer 1

Given,

Treatment1 : n1 = 75 , x1 = 47 , sample proportion = 47/75 = 0.6267

Treatment1 : n2 = 62 , x2 = 43 , sample proportion =43/62= 0.6935

The value of the pooled proportion is computed as,

= ( x1 + x2 ) / ( n1 + n2 ) = (47+43)/(75+62) = 90/137 = 0.6569

Null and alternative hypothesis -

Right tailed test.

Test statistics :

P-value :

P-value for this right tailed test is given by,

p-value = P( z > test statistics ) = P( z > -0.820 )

P( z > -0.820 ) = 1 - P( z <= -0.820 )

Using Excel function,   =NORMSDIST(z)

P( z <= -0.820 ) = NORMSDIST(-0.820 ) =0.206108

So, p-value = 1 - 0.206108 = 0.794

Decision rule :

Reject H0, if p-value less than significance level.

Here, p-value=0.794 is greater than significance level 2.5%, 5%, 10%

So fail to reject null hypothesis.

Correct option is , there is not a significant difference between the true proportions at the 10% level of significance