Question

A researcher asked four of his doctor friends if they thought the new Brand X lotion that he developed was effective. Three responded​ "yes." The researcher used the results to write a TV advertisement claiming that ​"3 out of 4 doctors prefer Brand X lotion​." Is this a valid​ inference? Explain.
Use pencil and paper. Explain all of the factors that are necessary for an inference to be valid.
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Part 1
Is the inference​ valid? Explain. Choose the correct answer below.
A.
​No, because 3 doctors said yes out of 7 who were asked.
B.
​No, because the doctors asked are not a representative sample of the population of doctors.
C.
​Yes, because 3 doctors said yes out of 4 who were asked.
D.
​Yes, because the doctors asked are a representative sample of the population of doctors.

Answer 1

Answer: B.

Explanation (concise):
- The statement "3 out of 4 doctors prefer Brand X" is true only for the four friends who were asked, but it is not a valid inference about doctors in general.
- For an inference about a population proportion to be valid you need:
1. A representative sample (usually obtained by random sampling from the target population).
2. Adequate sample size so sampling variability is not extreme.
3. Independence and unbiased data collection (no leading questions, no conflict of interest that could influence responses).
4. Appropriate measurement (the question asked actually measures “preference” fairly).
- The researcher asked four friends (nonrandom, highly likely biased) and n = 4 is far too small to estimate a population proportion reliably. Because the sample is not representative and is subject to selection bias (and potential researcher influence), the advertised claim is not justified.

If the researcher wants to generalize, he should survey a larger, randomly chosen sample of doctors (or run a controlled study) before making such a claim.