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21 January, 13:57

A marketing representative wants to estimate the proportion of people in a state who like the new design on the packaging of a certain cleaning product. The representative interviewed 100 people at a certain supermarket, and 82 people indicated that they liked the new design. Have the conditions for creating a confidence interval for the population proportion been met?

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  1. 21 January, 14:02
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    The conditions for creating a confidence interval for the population proportion have been met.

    Explanation:

    There are the following conditions to build a confidence interval for a population proportion:

    Sample of size n from a large population

    Individuals chosen independent of one another

    At least 15 failures and 15 sucesses in the sample.

    In this problem, we have that:

    Sample of 100 people

    They are chosen at random at the market, so it means that the probability that an individual likes the new design is independent of any other individuals.

    82 successes and 18 failures.

    So yes, the conditions for creating a confidence interval for the population proportion have been met.
  2. 21 January, 14:05
    0
    If the representative interviewed 100 people at a certain supermarket, and 82 people indicated that they liked the new design, the conditions for creating a confidence interval for the population proportion NOT been met because:

    the sample was from one source - supermarket. The number sampled (100 people) in comparison with the population of people in the state is negligible.

    Explanation:

    When constructing confidence intervals the assumptions and conditions of the central limit theorem must be met in order to use the normal model.

    Randomization Condition: The data must be sampled randomly. Is one of the good sampling methodologies discussed in the Sampling and Data chapter being used?

    Independence Assumption: The sample values must be independent of each other. This means that the occurrence of one event has no influence on the next event. Usually, if we know that people or items were selected randomly we can assume that the independence assumption is met.

    10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population.

    Sample Size Condition: The sample size must be sufficiently large.
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