3. Suppose we want to test the null hypothesis H0:p=0.20𝐻0:𝑝=0.20 against the alternative hypothesis Ha:p<0.20𝐻𝑎:𝑝<0.20 . Suppose also that we observed 25 successes in a random sample of 100 subjects and the level of significance is 0.05. What is the observed test statistic for this test?Multiple choice 1 Question 3 z=1.15𝑧=1.15 z=−1.25𝑧=−1.25 z=−1.15𝑧=−1.15 z=1.25
Question
- Suppose we want to test the null hypothesis H0:p=0.20𝐻0:𝑝=0.20 against the alternative hypothesis Ha:p<0.20𝐻𝑎:𝑝<0.20 . Suppose also that we observed 25 successes in a random sample of 100 subjects and the level of significance is 0.05. What is the observed test statistic for this test?Multiple choice 1 Question 3 z=1.15𝑧=1.15 z=−1.25𝑧=−1.25 z=−1.15𝑧=−1.15 z=1.25
Solution
To calculate the observed test statistic, we use the formula for the z-score:
z = (p̂ - p0) / sqrt((p0 * (1 - p0)) / n)
where:
- p̂ is the observed proportion of successes,
- p0 is the hypothesized proportion of successes under the null hypothesis,
- n is the sample size.
In this case:
- p̂ = 25 / 100 = 0.25,
- p0 = 0.20,
- n = 100.
Substituting these values into the formula, we get:
z = (0.25 - 0.20) / sqrt((0.20 * (1 - 0.20)) / 100) z = 0.05 / sqrt(0.016) z = 0.05 / 0.12649 z = 0.395
However, because we are testing the alternative hypothesis that p < 0.20, we take the negative of this z-score, so z = -0.395.
None of the provided options match this result. There may be a mistake in the problem or the provided options.
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