business statistics

Question

You wish to test the following claim (HaHa) at a significance level of α=0.001α=0.001.      Ho:p1=p2Ho:p1=p2 

      Ha:p1≠p2Ha:p1≠p2 You obtain 115 successes in a sample of size n1=375n1=375 from the first population. You obtain 87 successes in a sample of size n2=325n2=325 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.

What is the z-score of the critical value? (Report answer accurate to three decimal places.) 

±______

What is the standardized test statistic for this sample? (Report answer accurate to three decimal places.) 

_______

Answer 1

Hi Salma,

First, for your z-critical value:

alpha=0.001

We have a two-sided test, so we need alpha/s

alpha/2=0.0005

Subtract from 1 to get z-probability:

1-0.0005=0.9995

Look for this probability in the interior of the z-table; then look to the left column and top row for the z-score. There are a couple options, but go with the lowest one. This gives:

z= +/- 3.270

Now, let's get to your test statistic: we are assuming, under null hypothesis, that there is no difference in population means, so:

z=(p^₁ - p^₂ - 0)/sqrt[(p^(1-p^))((1/n₁) + (1/n₂))]

Now, breaking this down:

p^₁=first sample proportion

p^₂=second sample proportion

0=hypothesized difference under null hypothesis

p^=combined sample proportion

n₁=first sample size

n₂=second sample size

z=(p^₁ - p^₂ - 0)/sqrt[(p^(1-p^))((1/n₁) + (1/n₂))]

Now, let's identify these variables for our problem:

p^₁=115/375=0.307

p^₂=87/325=0.268

p^=(115+87)/(375+325)=0.289

n₁=375

n₂=325

Finally:

z=(p^₁ - p^₂ - 0)/sqrt[(p^(1-p^))((1/n₁) + (1/n₂))]

z=[(0.307-0.268)-0]/sqrt[(0.289*0.711)((1/375) + (1/325))]

z=1.135

That is your test statistic. I hope this helps.