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 α = 0.05 to test the following claim:

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?

What are the hypothesis for this test question?

NL (1) AL (2)
Sample Size 333 525
Below 3.5 124 241

c08.p082.q003

Question

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 α = 0.05 to test the following claim:

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?

What are the hypothesis for this test question?

NL (1)	AL (2)
Sample Size	333	525
Below 3.5	124	241

c08.p082.q003

BlackTom AI · Accepted Answer

Because the claim is that the proportions “differ,” the alternative hypothesis must be two‑sided: H1: p1 ≠ p2.

1. Relevant knowledge points and concepts
- p1 and p2 denote the population proportions of interest: let p1 = proportion of National League (NL) pitchers with ERA < 3.5 and p2 = proportion of American League (AL) pitchers with ERA < 3.5.
- For testing whether two independent proportions differ, the null hypothesis (H0) always states no difference (equality), and the alternative states the difference form suggested by the claim (two‑sided if the word “differs” is used).
- Null hypothesis is the default assumption to be tested; equality is required to compute pooled estimates if performing a z‑test.

2. Step‑by‑step reasoning
- The claim: “the proportion of NL pitchers with an ERA below 3.5 differs from the proportion of AL pitchers….” The word “differs” indicates a two‑tailed alternative.
- Therefore H1 must be p1 ≠ p2, and H0 must be p1 = p2.

3. Calculations (intermediate numerical checks)
- Sample proportions (for context): p̂1 = 124/333 ≈ 0.373, p̂2 = 241/525 ≈ 0.459.
- Observed difference p̂1 − p̂2 ≈ 0.373 − 0.459 ≈ −0.086 (this is the sample evidence; hypotheses remain symbolic as above).

4. Why the correct answer is correct
- H0: p1 = p2 expresses no difference between population proportions; H1: p1 ≠ p2 exactly matches the wording “differs” and sets up a two‑tailed test appropriate for α = 0.05.

5. Why other forms would be incorrect
- One‑sided alternatives (p1 > p2 or p1 < p2) would be incorrect because the claim does not specify a direction.
- Writing H0 as an inequality (e.g., p1 ≥ p2) would be incorrect because null hypotheses for difference tests are stated as equality.

Conclusion: The correct hypotheses are H0: p1 = p2 and H1: p1 ≠ p2, which matches the provided answer.

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