89` Advanced StatisticsQuiz#105/27/19

  1. (7.5%) Suppose that P(A|B) = 0.2, P(A|B`) = 0.3, and P(B) = 0.8. What are P(A), P(AB), and P(AB)?
  1. (7.5%) Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products received good reviews, 60% of moderately successful products received good reviews, and 10% of poor products received good reviews. In addition, 40% of the products have been highly successful, 35% of the products have been moderately successful, and 25% of the products have been poor successful.
  1. What is the probability that a product attains a good review?
  2. If a new design attains a good review, what is the probability that it will be a highly successful product?
  3. If a new design does not attains a good review, what is the probability that it will be a highly successful product?
  1. (10%) The following table lists the history of 940 orders for features in an entry-level computer product.

Extra Memory

No / Yes
Optional High Speed Processor /

No

/ 514 / 68
Yes / 112 / 246

Let A be the event that an order requests the optional high-speed processor, and let B be the event that an order requests extra memory. Determine the following probabilities.

  1. P(AB), P(AB), P(A`B), and P(A`B`).
  2. What is the probability that an order requests an optional high-speed processor given that the order requests extra memory?
  3. What is the probability that an order requests extra memory given that the order requests an optional high-speed processor?
  1. (15%) Show that, for the simple linear regression model, the following statements are true:
  1. b. c.
  1. (15%) Suppose that we are interested in fitting a simple linear regression model , where the intercept, , is known.
  1. Find the least squares estimator of .
  2. What is the variance of the estimator of the slop in part (a)?
  3. Find the expression for a 100(1-)% C.I. for the slop . Is this interval longer than the corresponding interval for the case where both the intercept and slop are known?
  1. (15%) A random sample of n = 25 observations was made on the time to failure of an electronic component and the temperature in the application environment in which the component was used.
  1. Given that r = 0.83, test the hypothesis that = 0 , using  = 0.05. What is the P-value for this test?
  2. Find a 95% confidence interval on .
  3. Test the hypothesis H0: = 0.8 versus H1:  0.8 using  = 0.05. What is the P-value for this test?
  1. (30%) Data on compressive strength x and intrinsic permeability y of various concrete mixes and cures are presented. Summary quantities are n = 14, Assume that the two variables are related according to the simple linear regression model.
  1. Calculate the least squares estimates of the slope and intercept.
  2. Use the equation of the fitted line to predict what permeability would be observed when the compressive strength is x = 4.3.
  3. Given a point estimate of the mean permeability when compressive strength is x = 3.7.
  4. Suppose that the observed value of permeability at x = 3.7 is y = 46.1. Calculate the value of the corresponding residual.
  5. Test for the significance of regression using = 0.05. Find the P-value for the this test? What do you conclude?
  6. Estimate 2 and the standard deviation of the estimated slope.
  7. What is the standard error of the intercept in this model?
  8. Find the 95% C.I.s on the slope, intercept, mean permeability when x = 2.5.