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**University of Toronto Department of Materials Science and Engineering Engineering Statistics MSE 238h1s FOURTH QUIZ , 10 th FEBRUARY 2005, 14:15-14:45 QUESTION 1 ( MARKS 30 ) **

Let P(A) = 0.4 and P(A B) = 0.7 What is P(B) if (a) A and B are independent P(A B)= P(A)P(B) P(A B)=P(A)+P(B)-P(A B) 0.7=0.4+P(B)-0.4.P(B) P(B)=0.5 (b) A and B mutually exclusive P(A B)=0 P(A B)=P(A)+P(B)-P(A B) 0.7=0.4+P(B) P(B)=0.3

**QUESTION 2 ( MARKS 35 ) **

Find the probabilities of getting (a) eight heads in a row with balanced coin (1/2) 8 (b) three 3’s and then a 4 or a 5 in four rolls of a balanced die (1/6) 3 .(2/6) (c) five multiple-choice questions answered correctly, if for each question the probability of answering it correctly is 1/3 (1/3) 5

**QUESTION 3 ( MARKS 35 ) **

A company is analyzing the compressive strength of a new material. Compressive strength is normally distributed with 2 = 1000(psi) 2 . A random sample of 12 specimens has a mean compressive strength of

*x*

3250

*psi*

(a) Construct a 95% two-sided confidence interval on mean compressive strength. Z /2 = Z 0.025

= 1.96 3250-1.96(1000/12) 0.5

3250+1.96(1000/12) 0.5

) 3232.11

3267.89 (b) Construct a 99% two-sided confidence interval on mean compressive strength. Compare the width of this confidence interval with the width of the one found in part (a). Z /2 = Z 0.005

= 2.58 3250-2.58(1000/12) 0.5

3250+2.58(1000/12) 0.5

3226.5

3273.5 This interval is longer because it is calculated with a higher level of confidence.