Final answer:
The 95% confidence interval estimate for the proportion of all married men that prefer Miyagi Instant Coffee is between 12.16% and 27.84%. This does not imply that 95% of married men prefer Miyagi, but that there is high confidence that the true proportion falls within this range.
Step-by-step explanation:
To estimate the proportion of all married men who prefer Miyagi Instant Coffee over the major brands with a 95% confidence interval, the proportion from the sample is used as a starting point. In the survey, 20 out of 100 married men preferred Miyagi, indicating a sample proportion of 0.20 (or 20%).
To calculate the 95% confidence interval, we use the formula for the interval of a proportion:
Confidence Interval = p ± z*sqrt((p(1-p)/n)),
where:
- p is the sample proportion (0.20),
- z is the z-score corresponding to the desired confidence level (1.96 for 95%), and
- n is the sample size (100).
Using this information, the confidence interval is calculated as follows:
Lower limit = 0.20 - 1.96*sqrt((0.20*0.80)/100) = 0.20 - 0.0784 = 0.1216
Upper limit = 0.20 + 1.96*sqrt((0.20*0.80)/100) = 0.20 + 0.0784 = 0.2784
Thus, the 95% confidence interval estimate for the true proportion of all married men who prefer Miyagi over the other brands is between 12.16% and 27.84%.
We interpret this interval to mean that we are 95% confident that the true proportion of all married men who prefer the Miyagi brand falls within this range. It does not mean that 95% of married men prefer the brand, nor that the interval has a 90% confidence level (as this would require a different z-score), nor that the survey's sample size was 200, or that it was conducted among women (which is unrelated to the calculation).