I just cant figure this out Please help The table shows the

I just can\'t figure this out. Please help!!

The table shows the number of pitchers with E.R.A\'s below 3.5 in a random sample of sixty pitchers from the National League and in a random sample of fifty-two pitchers from the American League. Assume that you plan to use a significance level of = 0.05 to test the claim that p1 p2. Find the critical value(s) for this hypothesis test. Do the data support the claim that the proportion of National League pitchers with an E.R.A. below 3.5 differs from the proportion of American League pitchers with an E.R.A. below 3.5?

National League American League

Number of pitchers in sample: 60 52

Number of pitchers with E.R.A. below 3.5: 16 10

Answers: A) z = ± 2.575; no B) z = ± 1.96; no C) z = 1.645; yes D) z = ± 1.645; yes

Solution

The table shows the number of pitchers with E.R.A\'s below 3.5 in a random sample of sixty pitchers from the National League and in a random sample of fifty-two pitchers from the American League.

level of significance (alpha) = 0.05

Hypothesis for the test is,

H0 : p1 = p2 Vs H1 : p1 p2

where p1 is the proportion of National League pitchers with an E.R.A. below 3.5

p2 is the proportion of American League pitchers with an E.R.A. below 3.5

Here the test is two sided.

x1 = Number of pitchers with E.R.A. below 3.5 (national league) = 16

n1 = Number of pitchers in sample (national league) = 60

x2 = Number of pitchers with E.R.A. below 3.5 (American league) = 10

n2 = Number of pitchers in sample (American league) = 52

p1^ = x1/n1 = 16 / 60 = 0.2667

p2^ = x2 / n2 = 10 / 52 = 0.1923

The pooled estimate of proportion is,

p^ = (x1 + x2) / (n1 + n2)

= (16 + 10) / (60 + 52) = 0.2321

q^ = 1 - p^ = 1 - 0.2321 = 0.7679

The test statistic for testing p1 and p2 is,

z = (p1^ - p2^) / sqrt [ p^q^/n1 + p^q^/n2 ]

z = (0.2667 - 0.1923) / sqrt [ (0.2321*0.7619) / 60 + (0.2321*0.7619)/52 ]

z = 0.9295

The critical value is,

+ or - 1.96

The test statistic value < critical value

Fail to reject H0 at 5% level of significance.

And conclude that the proportion of National League pitchers with an E.R.A. below 3.5 is same (no difference) as the proportion of American League pitchers with an E.R.A. below 3.5.