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    Type 2 Error

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    1. In random samples of 25 from each of two normal populations, we found the following statistics:
    x1 bar = 524 s1 = 129
    x2 bar = 469 s2 = 131
    a) estimate the difference between the two population means with 95% confidence.
    b) repeat part a) increasing the standard deviations of s 1 to 255 and s 2 to 260.
    c) describe what happens when the sample standard deviations get larger.
    d) repeat part a) with samples of size 100.
    e) discuss the effects of increasing the sample size.

    2. Calculate the following:
    a) Determine the sample size necessary to estimate a population mean to within 1 with a 90% confidence given that the population standard deviation is 10.

    b) Suppose that the sample was calculated at 150. Estimate the population mean with 90% confidence.

    3. a) Calculate the probability of a type ii error for the following hypotheses when u = 37:
    ho: u = 40
    h1: u < 40
    the significance level is 5%, the population standard deviation is 5, and the sample size is 25.
    b) repeat part (a) with a significance level = 15%.
    c) What is the effect of increasing the significant level on the type ii error.

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    Solution Preview

    See the attached file.

    1. a) difference=524-469=55.
    the critical value for 95% confidence interval is 2.01 (TINV(0.05,48), 48 is the degree of freedom).
    standard error=sqrt(129^2/25+131^2/25)=36.771
    margin of error=2.01*36.771=73.91
    upper limit: 55+73.91=128.91
    lower limit: 55-73.91=-18.91
    therefore, the 95% confidence interval for the difference is [-18.91, 128.91].

    b) standard error=sqrt(255^2/25+260^2/25)=72.835
    margin of error=2.01*72.835=146.40
    upper limit: ...

    Solution Summary

    The solution discusses the type 2 error in the given statistics problems.