Question

Conduct a test at the a = 0.05 level of significance by determining (a) the null and alternative hypotheses (b) the test statistic (c) the critical value, and (d) the P-value. Assume that the samples were obtained independently using simple random sampling.

1. Test whether p₁ is not equal to p₂. Sample data: x₁ = 804, n₁ = 874, x₂ = 902, n₂ = 954

Answer 1

1. Test whether p1 is not equal to p2. Sample data: x1 = 804, n1 = 874, x2 = 902, n2 = 954

a) Null hypothesis :Ho : p1 = p2

Alterrnative Hypothesis: Ha: p1 p2

=0.05

Two tailed test.

b) Method 1 : individual proportions

: : Sample Propotion of Sample 1 = x1/n1 = 804/874 = 0.9199

: Sample Propotion of Sample 2 = x2/n2 = 902/954 = 0.9455

c) Critical value :

For two tailed test : Critical value = Z/2 = Z0.025 = 1.96

As Value of the test statistic Z is less than Critical Value i.e. ( -2.1767<-1.96 ); Reject Null Hypothesis

d)p-value

For two tailed test :

p(Z<-2.1767) = 0.0148

p-value = 2 x 0.0148 = 0.0296

As P-Value i.e. is less than Level of significance i.e (P-value:0.0296 < 0.05:Level of significance); Reject Null Hypothesis

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a) Null hypothesis :Ho : p1 = p2

Alterrnative Hypothesis: Ha: p1 p2

=0.05

Two tailed test.

b) Method 2 : pooled proportions

: : Sample Propotion of Sample 1 = x1/n1 = 804/874 = 0.9199

: Sample Propotion of Sample 2 = x2/n2 = 902/954 = 0.9455

c) Critical value :

For two tailed test : Critical value = Z/2 = Z0.025 = 1.96

As Value of the test statistic Z is less than Critical Value i.e. ( -2.1767<-1.96 ); Reject Null Hypothesis

d)p-value

For two tailed test :

p(Z<-2.1913) = 0.0142

p-value = 2 x 0.0142 = 0.0284

As P-Value i.e. is less than Level of significance i.e (P-value:0.0284 < 0.05:Level of significance); Reject Null Hypothesis