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I have this question:

It is reported that 45% of the population prefer apples and 55% prefer oranges. Describe the methodology for conducting the survey.

I am not sure if my reasoning is correct:

Assuming the population prefers either apples or oranges (not both nor neither)

  1. Do a sample of the population, assign a value of 1 to apple, 0 to oranges (for example)
  2. Take the sample mean and sample variance
  3. Plot against the t-distribution
  4. Set a confidence interval
  5. Compare if the sample mean lies within the confidence interval of the t-value corresponding to 45% (apple).
  6. If so, conclude that the statement is true, for the given confidence interval.

Sorry a bit lengthy, but thanks for the help.

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Before I can answer your question, I have a few questions to ask you first:

  • It seems like those numbered list are "your" methodology for
    conducting the survey, right?
  • Is that the whole question or just a part of question?

It is reported that 45% of the population prefer apples and 55% prefer oranges. Describe the methodology for conducting the survey.

  1. You probably want to elaborate this part of your answer since the question is asking for the survey methodology. Just saying, "Do a sample of the population" seems too vague.

    In terms of survey sampling, you would need to understand how survey methodologies and techniques should be implemented in your case (e.g. sample size, survey sampling methods, etc.). There are several ways to do survey sampling. But, there isn't just enough information to help you here.

  2. I don't know how you could calculate the sample means... you can calculate the predicted proportion of people who prefer apples ($\hat p$) and obviously $(1-\hat p)$ is the predicted proportional of people who prefer oranges. Based on these result, you can construct a simple binomial proportion confidence interval.

  3. I am not sure if this is even possible with proportions.

  4. Refer 2.

  5. Addition to your answer, you could even conduct a hypothesis test. Make sure the conditions are met in order to proceed with this test.

  6. Make a conclusion based on the results.

Under the assumption that the samples are independent, you could conduct a hypothesis test for the difference between proportions but it seems unnecessary based on the question.

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It would be inconvenient for all concerned to cast this in terms of means. You'd be better off comparing your sample proportion to the population proportion (choose one: the one for apples or the one for oranges). You can do this after calculating the standard error of your sample proportion: see wikipedia, below the heading labelled "Calculations assuming random sampling." Once you have that standard error, you can create a confidence interval for the population parameter and/or conduct a significance test for the difference between the sample proportion and the one hypothesized to be true for the population.

I agree with @Ken about the need to specify more about your sampling plan.

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