Suppose we are interested in the proportion of people who prefer Lays chips to a store brand chip. In particular, we want to test whether this proportion is greater than 0.5. In a taste test based on a random sample of 50 people, 29 of them preferred Lays. Is this compelling evidence that the population proportion is actually greater than 0.5?

Let
𝑝
be the population proportion of people who prefer Lays. What are the appropriate hypotheses for the test?

Question

Suppose we are interested in the proportion of people who prefer Lays chips to a store brand chip.  In particular, we want to test whether this proportion is greater than 0.5.  In a taste test based on a random sample of 50 people, 29 of them preferred Lays.  Is this compelling evidence that the population proportion is actually greater than 0.5?

Let 
𝑝
 be the population proportion of people who prefer Lays.  What are the appropriate hypotheses for the test?

BlackTom AI · Accepted Answer

When testing whether a population proportion exceeds 0.5, the null hypothesis fixes the proportion at 0.5 and the alternative is one-sided to the right: H0: p = 0.5 versus HA: p > 0.5.

1. Relevant knowledge points and concepts
- Hypothesis testing for a population proportion uses H0 to state the status-quo value (here 0.5) and HA to state the research claim. 
- Because the question asks whether the proportion is greater than 0.5, the alternative must be one-sided (greater-than). 
- Check normal-approximation conditions: np0 and n(1−p0) should be at least about 10 to use the z-test approximation.

2. Complete reasoning process step by step
- The parameter of interest is p, the population proportion preferring Lays.
- The claim is "greater than 0.5," so the alternative hypothesis must be p > 0.5.
- The null hypothesis is the complement (equality) H0: p = 0.5.

3. Calculations and intermediate steps
- Sample size n = 50, number preferring Lays = 29.
- Sample proportion p̂ = 29/50 = 0.58.
- For checking approximation: np0 = 50(0.5) = 25 and n(1−p0) = 50(0.5) = 25, both ≥ 10, so a normal-approximation z-test would be appropriate if performing the test.

4. Why the correct answer is correct
- The sample proportion (0.58) is above 0.5, and the research question explicitly asks if the population proportion is greater than 0.5, so the alternative must be p > 0.5 with null p = 0.5. This matches the one-sided right-tail hypothesis pair.

5. Why the other options are incorrect
- H0: p = 0.5 versus HA: p < 0.5 is the wrong direction; it would test for fewer than half preferring Lays, opposite the stated interest.
- H0: p = 0.5 versus HA: p ≠ 0.5 is a two-sided test; it tests for any difference from 0.5 but does not specifically test the claim that p is greater than 0.5.

6. Conclusion
- Therefore the appropriate hypotheses are H0: p = 0.5 versus HA: p > 0.5, which matches the provided answer.

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