[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.

类似问题

Suppose we are performing a Proportion Test to find out the percentage of international students in a class. Let p = the proportion of international students. The null hypothesis H0 is : p = 0.3. We take a sample of 10, and the observed value is 2. What is the test statistic?

When evaluating a new project, firms should include in the projected cash flows all of the following EXCEPT:

Provide the best rationale for what's wrong with supply chains today?

What is Blockchain and how is it relevant to supply chain management?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单