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2 November, 03:00

For a recent report on sleep deprivation, the Centers for Disease Control and Prevention interviewed 11508 residents of California and 4860 residents of Oregon. In California, 932 respondents reported getting insufficient rest or sleep during each of the preceding 30 days, while 452 of the respondents from Oregon reported the same.

Calculate a 95 % confidence interval for the difference between the proportions of Californians and Oregonians, pCalifornia-pOregonpCalifornia-pOregon, who report getting insufficient rest. Round your answer to 4 decimal places.

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  1. 2 November, 03:10
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    95% confidence interval for the difference between the proportions of Californians and Oregonians is (-0.0070, - 0.0130).

    Step-by-step explanation:

    Confidence interval for a proportion is given as p + / - margin of error (E)

    Californians

    p = 932/11508 = 0.08

    n = 11508

    C = 95% = 0.95

    Significance level = 1-C = 1-0.95 = 0.05 = 5%

    critical value corresponding to infinity degrees of freedom and 5% significance level is 1.96

    E = critical value * sqrt[p (1-p) : n] = 1.96 * sqrt[0.08 (1-0.08) : 11508] = 1.96 * 0.00253 = 0.0050

    Lower limit = p - E = 0.08 - 0.0050 = 0.0750

    Upper limit = p + E = 0.08 + 0.0050 = 0.0850

    Oregonians

    p = 452/4860 = 0.09

    n = 4860

    critical value = 1.96

    E = 1.96 * sqrt[0.09 (1-0.09) : 4860] = 1.96 * 0.0041 = 0.0080

    Lower limit = p - E = 0.09 - 0.0080 = 0.0820

    Upper limit = p + E = 0.09 + 0.0080 = 0.0980

    Difference in lower limit of proportion = 0.0750 - 0.0820 = - 0.0070

    Difference in upper limit of proportion = 0.0850 - 0.0980 = - 0.0130

    95% confidence interval for the difference in proportion is between a lower limit of - 0.0070 and an upper limit of - 0.0130.
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