The GO transportation system of buses and commuter trains operates on the honor system. Train travelers are expected to buy their tickets be

Question

The GO transportation system of buses and commuter trains operates on the honor system. Train travelers are expected to buy their tickets before boarding the train. Only a small number of people will be checked on the train to see whether they bought a ticket. Suppose that a random sample of 200 train travelers was sampled and 24 of them had failed to buy a ticket. Estimate with 90% confidence the proportion of all train travelers who do not buy a ticket. Interpret the confidence interval you found.

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4 days 2021-10-08T11:05:00+00:00 1 Answer 0

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    2021-10-08T11:06:10+00:00

    Answer:

    The 90% confidence interval for the population proportion of all train travelers who do not buy a ticket is (0.08, 0.16).

    Step-by-step explanation:

    The (1 – α)% confidence interval for the population proportion is:

    CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}

    The information provided is:

    n = 200

    X = 24

    Confidence level = 90%

    Compute the value of sample proportion as follows:

    \hat p=\frac{X}{n}=\frac{24}{200}=0.12

    Compute the critical value of z for 90% confidence level as follows:

    z_{\alpha/2}=z_{0.10/2}=z_{0.05}=1.645

    Compute the 90% confidence interval for the population proportion of all train travelers who do not buy a ticket as follows:

    CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}

         =0.12\pm 1.645\times\sqrt{\frac{0.12(1-0.12)}{200}}\\\\=0.12\pm0.038\\=(0.082, 0.158)\\\approx (0.08, 0.16)

    The 90% confidence interval for the population proportion of all train travelers who do not buy a ticket is (0.08, 0.16).

    The 90% confidence interval (0.08, 0.16) for the population proportion of all train travelers who do not buy a ticket implies that there is a 0.90 probability that the true proportion lies in this interval.

    Or if 100 such intervals are computed then 90 of those intervals will consist of the true proportion.

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