75.1k views
0 votes
The marketing research department of an instant-coffee company conducted a survey of married men to determine the proportion who prefer their brand, Miyagi Instant Coffee, to the major brands on the market. A random sample of 100 married men was gathered, and twenty of those men preferred the company's brand to the alternatives in the survey. Use a 95% confidence interval to estimate the proportion of all married men who prefer Miyagi over the major instant coffee brands. Be sure to interpret your answer.

A) 20% of married men prefer Miyagi Instant Coffee
B) The confidence interval is 90%
C) The sample size is 200
D) The survey was conducted among women

1 Answer

3 votes

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).

User Wilkas
by
7.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories