[Same setup as Questions 1 & 2:] 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?

What is the p-value?

Question

[Same setup as Questions 1 & 2:]  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?

What is the p-value?

BlackTom AI · Accepted Answer

Compute the one-sided z-test statistic for a proportion: z = (p̂ − p0) / sqrt(p0(1−p0)/n).

1) Relevant knowledge points and concepts
- Hypotheses: H0: p = 0.5 vs. Ha: p > 0.5 (one-sided, upper-tail test).
- Use the one-sample z-test for a proportion when np0 and n(1−p0) are both at least 10 (normal approximation).
- p-value for an upper-tail test is P(Z ≥ z_obs) where Z ~ N(0,1).

2) Complete reasoning and setup
- Sample size n = 50, successes x = 29, so sample proportion p̂ = 29/50.
- Check normal approximation: np0 = 50(0.5) = 25 ≥ 10 and n(1−p0) = 25 ≥ 10, so approximation is valid.

3) Calculations (all intermediate steps)
- p̂ = 29/50 = 0.58.
- Standard error under H0: SE = sqrt[p0(1−p0)/n] = sqrt[0.5×0.5 / 50] = sqrt[0.25 / 50] = sqrt[0.005] ≈ 0.0707107.
- z = (p̂ − p0) / SE = (0.58 − 0.5) / 0.0707107 = 0.08 / 0.0707107 ≈ 1.13137.
- p-value (upper-tail) = P(Z ≥ 1.13137) = 1 − Φ(1.13137) ≈ 1 − 0.8708 ≈ 0.1292 ≈ 0.129.

4) Why the correct answer is correct
The calculation follows the standard one-proportion z-test procedure for a one-sided alternative. The observed z ≈ 1.13 corresponds to an upper-tail probability of about 0.129, so the p-value is 0.129. This directly matches the provided correct answer.

5) Why the other options are incorrect
- 0.258 is approximately double 0.129 and would be the two-sided p-value (not appropriate for the one-sided Ha: p > 0.5).
- 0.742 and 0.871 are complements or incorrect table values (e.g., Φ(1.13) ≈ 0.871 is the lower-tail probability; 0.871 is not the p-value for an upper-tail test).

6) Conclusion
The one-sided p-value is approximately 0.129, confirming the provided answer 0.129.

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