A researcher is interested to determine the average age at which people obtain their first credit card. If past information shows a mean of

Question

A researcher is interested to determine the average age at which people obtain their first credit card. If past information shows a mean of 22 years and a standard deviation of 2 years, what size sample should be taken so that at 95% confidence the margin of error will be 3 months or less?

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Aaliyah 2 weeks 2022-01-08T20:47:38+00:00 1 Answer 0 views 0

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    2022-01-08T20:49:26+00:00

    Answer:

    The sample size should be of at least 246.

    Step-by-step explanation:

    We have that to find our \alpha level, that is the subtraction of 1 by the confidence interval divided by 2. So:

    \alpha = \frac{1-0.95}{2} = 0.025

    Now, we have to find z in the Ztable as such z has a pvalue of 1-\alpha.

    So it is z with a pvalue of 1-0.025 = 0.975, so z = 1.96

    The margin of error M is

    M = z*\frac{\sigma}{\sqrt{n}}

    In which \sigma is the standard deviation of the population and n is the size of the sample.

    In this problem, we have that:

    \sigma = 2

    We want the margin of error to be of 3 months. However the standard deviation is in years. So the margin of error must be in years. So M = 3/12 = 0.25.

    As n increases, the margin of error decreases. So we need a sample of size at least n when M = 0.25.

    M = z*\frac{\sigma}{\sqrt{n}}

    0.25 = 1.96*\frac{2}{\sqrt{n}}

    0.25\sqrt{n} = 3.92

    \sqrt{n} = 15.68

    \sqrt{n}^{2} = (15.68)^{2}

    n = 246

    The sample size should be of at least 246.

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