Spring 2008

Equally weighted problems. Your best problem counts double.

1. Let X and Y have a bivariate normal distribution with m_X=12, m_Y=64, s_X²=25, s_Y²=36, and the correlation coefficient r_XY=0.7.

(a) Compute P(X ³ 16).

(b) Compute P(X ³ 16 | Y=50).

2. Let X be N(m,9), and (X₁,X₂,X₃,X₄,X₅) be a random sample of size 5 of X.

(a) Find the bias of T₁ as an estimator of m, if

T1= 1 10 X1 + 2 5 X3 + 1 5 X4.

(b) Determine whether T₂ or T₃ is the more efficient estimator of m, where

T2= 1 5 X1 + 3 5 X2 + 1 5 X4        or        T3= 1 16 X2 + 1 16 X3 + 3 4 X4+ 1 8 X5.

3. Let X be U(b-[3/4],b+2). Find the method of moments estimator for b.

4. A population has a density function given by

f(x)= 1 6q4 x3e-x/q

for x > 0, where q > 0 is unknown. Use calculus methods to find the maximum likelihood estimator of q, based on a random sample of size n.

5. Researchers studying commercial air-filtering systems for noise pollution reported the following sample results for mean noise level. Construct a 95% confidence interval for the difference in the mean noise levels, assuming that the populations are normally distributed with equal variances.

Unit A:       n=6       [`x]=91.3       s=1.1 Unit B:       n=8       [`x]=87.5       s=3.7

6. From past experience it is known that the mean compressive strength (measured in units of 100 pounds per square inch) of bricks is normally distributed with unknown mean m and known variance s²=36. The bricks will not be acceptable unless we can be reasonably sure that m > 50. To test the null hypothesis H₀: m = 50 against the alternative hypothesis H₁:m > 50, nine bricks will be randomly selected and their compressive strengths recorded. The null hypothesis will be rejected if [`x] > 53.3, where [`x] is the observed mean of the sample.

(a) Find the power function K(m) for this test.

(b) What is the significance level of this test?

7. An ice cream supplier claims that among the four most popular flavors, customers have these preference rates: 65% prefer vanilla, 15% prefer chocolate, 12% prefer neapolitan, and 8% prefer vanilla fudge. A random sample of 200 customers produces the results below. At the 0.05 significance level, test the claim that the percentages given by the supplier are correct.

Flavor       Vanilla       Chocolate       Neapolitan       Vanilla Fudge    Number prefer       120       40       18       22

1. Hypotheses

H₀:

H₁:

2. Test Statistic

3. Decision Rule (a =       )

Picture:

Reject H₀ if

Otherwise, fail to reject H₀.

4. Observed Value

5. Conclusion

Reject H₀/Fail to reject H₀ (Circle one)

In English:

8. Test the claim that more than one-fourth of all white collar criminals have attended college. U.S. Bureau of Justice statistics reveal that of a sample of 1400 randomly selected white collar criminals, 33% had attended college. Use a 0.05  level of significance.

1. Hypotheses

H₀:

H₁:

2. Test Statistic

3. Decision Rule (a =       )

Picture:

Reject H₀ if

Otherwise, fail to reject H₀.

4. Observed Value

5. Conclusion

Reject H₀/Fail to reject H₀ (Circle one)

In English: