It has been claimed that 68% of dog owners have driven a car with their dog sitting in their lap. Believing this percentage is too high, a researcher surveys a random sample of 260 dog owners and finds that 169 of these dog owners have let their dog sit in their lap while driving. If a hypothesis test is conducted at
𝛼
= 0.05, what should the researcher conclude?

Question

It has been claimed that 68% of dog owners have driven a car with their dog sitting in their lap.  Believing this percentage is too high, a researcher surveys a random sample of 260 dog owners and finds that 169 of these dog owners have let their dog sit in their lap while driving.   If a hypothesis test is conducted at 
𝛼
 = 0.05, what should the researcher conclude?

BlackTom AI · Accepted Answer

For testing a population proportion use a z-test: H0: p = 0.68 and Ha: p < 0.68 (the researcher believes 68% is too high, so the alternative is a left-tailed test).

1. Relevant knowledge points and concepts
- For proportions use p̂ = x/n. Under H0, the sampling distribution of p̂ is approximately normal when np0 and n(1−p0) are ≥ 10. The test statistic is z = (p̂ − p0)/√[p0(1−p0)/n]. For a left-tailed test the p-value = P(Z ≤ observed z).

2. Step-by-step reasoning
- Compute the sample proportion, verify normal approximation, compute standard error, z statistic, p-value, and compare to α = 0.05.

3. Calculations (all intermediate steps)
- n = 260, x = 169 → p̂ = 169/260 = 0.65.
- Check conditions: np0 = 260(0.68) = 176.8 ≥ 10; n(1−p0) = 260(0.32) = 83.2 ≥ 10. Normal approximation valid.
- Standard error under H0: SE = sqrt[p0(1−p0)/n] = sqrt[0.68(0.32)/260] = sqrt[0.2176/260] = sqrt[0.000836923] ≈ 0.02893.
- z = (p̂ − p0)/SE = (0.65 − 0.68)/0.02893 = −0.03/0.02893 ≈ −1.038.
- p-value = P(Z ≤ −1.038) ≈ 0.15 (about 0.149–0.150), which is greater than α = 0.05.

4. Why the correct answer is correct
- Because the p-value (~0.15) exceeds 0.05, we do not have sufficient evidence to reject H0 in favor of p < 0.68. The observed 0.03 difference is not large enough, relative to sampling variability, to be statistically significant at the 5% level.

5. Why the other options are incorrect
- Options claiming the p-value is smaller than 0.05 are wrong because the computed p-value is ~0.15. The option that says “p-value larger than 0.05, there is evidence against H0” is wrong because a larger p-value means insufficient evidence. The reasoning that compares the numeric difference 0.03 directly to 0.05 is incorrect: α is a significance level for the p-value, not a threshold for the raw difference.

6. Conclusion
- Final conclusion: Because the resulting P-value is larger than 0.05, there is not enough evidence against the null hypothesis. This matches the provided answer.

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