posted by sandy

Suppose we take a random sample of size n from a normal population with variance, σ2 . It can be shown that (n−1)s2/σ2 has a chi-square distribution with n−1 degrees of freedom, where s is the sample variance. Below is a random sample of size 8 drawn from a normal population. Use the above fact to compute a 95% confidence interval for the population variance, σ2 .
Sample 1: 11.6, 17.2, 15.0, 16.3, 22.9, 13.5, 16.4, 16.1

Below is a second random sample, independent from the first, of size 8 from a second normal population. Remembering that the F distribution is a ratio of independent chi- squares divided by their degrees of freedom, it can be shown that, under random, independent sampling, if the variances of the populations are equal, then s21/s2 has an F distribution with, in this case, 7 numerator and 7 denominator degrees of freedom (where the degrees of freedom are n − 1 for the corresponding samples). Test at α = .05 the null hypothesis that the variances are equal against the alternative that the variance of the first population is greater.
Sample 2: 17.7, 11.0, 17.0, 12.4, 10.8, 9.9, 17.2, 10.1

  1. MathGuru

    For your first problem:
    Standard Deviation = 3.29 (Standard deviation is the square root of the variance)
    Variance = 10.82 (Variance is standard deviation squared)

    Using a chi-square table for the endpoints:
    (n-1)s^2/16 to (n-1)s^2/1.69
    (8-1)10.82/16 to (8-1)10.82/1.69
    7(10.82)/16 to 7(10.82)/1.69
    75.74/16 to 75.74/1.69
    4.73 to 44.82 -->confidence interval for the variance


    For the second problem:
    Standard Deviation = 3.43
    Variance = 11.76

    Sample 1: n = 8; variance = 10.82; df = n - 1 = 7
    Sample 2: n = 8; variance = 11.76; df = n - 1 = 7

    Test statistic = sample 1 variance / sample 2 variance

    You can use the F-distribution at .05 level using the above information for degrees of freedom. This will be your critical value to compare to the test statistic. If the test statistic exceeds the critical value from the table, the null will be rejected in favor of the alternative hypothesis. If the test statistic does not exceed the critical value from the table, then the null is not rejected.

    I'll let you take it from here to finish. Check these calculations!

Respond to this Question

First Name

Your Answer

Similar Questions

  1. Statistics

    Suppose X1;X2;...;X5 is a random sample from a n(0; variance) distribution. define U = X1+X2+X3+X4; V = (X2)^2+(X3)^2+(X4)^2+(X5)^2 and W = U/sqrtV. (a) Name the distribution of U/4 and V/(variance) and give the values of their parameters. …
  2. Statistics

    Suppose that the fit to the simple linear regression of Y on X from 6 observations produces the following residuals: -3.3, 2.1, -4.0, -1.5, 5.1, 1.6. a) What is the estimate of sigma squared?
  3. Stor

    Here is a simple way to create a random variable X that has mean μ and stan- dard deviation σ: X takes only the two values μ−σ and μ+σ, eachwith probability 0.5. Use the definition of the mean and …
  4. Quantum Physics

    The variance σ2X=⟨(X^−⟨X^⟩)2⟩ of an operator, X^, is a measure of how large a range its possible values are spread over (the standard deviation is given by σ=σ2−−√). …
  5. statistics

    Suppose x has a normal distribution with mean μ = 35 and standard deviation σ = 9. (a) Describe the distribution of x values for sample size n = 4. (Use 2 decimal places.) μx = σx = (b) Describe the distribution …
  6. statistics

    Suppose a random sample of size 50 is selected from a population with σ = 10. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate). a. The …
  7. probability

    Let X be a continuous random variable. We know that it takes values between 0 and 3, but we do not know its distribution or its mean and variance. We are interested in estimating the mean of X, which we denote by h. We will use 1.5 …
  8. Statistics

    1.A sample of 25 bottles is taken from the production line at a local bottling plant. Assume that the fill amounts follow a normal distribution. The probability is 90% that the sample variance is less than what percent of the population …
  9. Statistics

    Suppose you have a sample of 6 observations from a normal population. The sample variance is equal to 4. Find a 90% lower confidence interval for the variance of the population.
  10. Statistics

    You are interested in estimating the mean of a population. you plan to take a random sample from the population and use the samples mean as an estimate of population mean. Assuming that the population from which you select your sample …

More Similar Questions