ENGI 3423 Probability and Statistics

Faculty of Engineering and Applied Science
2007 Fall

Final Examination Questions

From a box of seven identical valves, three valves are to be selected; one to be placed in an intake pipe, one in an outflow pipe and one in a drainage pipe. In how many different ways can these three valves be chosen?

[10]

From prior experience, it is believed that the mean potential of a certain type of battery is 12.0 V. The strength of this belief is represented by the standard deviation . A random sample of 81 such batteries has a sample mean potential of 12.2 V with a sample standard deviation of 0.3 V.
1. Construct a Bayesian 95% confidence interval estimate for the true mean potential µ.
  [8]
2. Construct a classical 95% confidence interval estimate for the true mean µ.
  [5]
3. Is there sufficient evidence to conclude that the true mean potential µ is not 12.0 V?
  [2]

The diameter of a cylinder is known to be a normally distributed random quantity with a mean and a standard deviation The diameter of a piston is known to be a normally distributed random quantity with a mean and a standard deviation .
1. Show that the probability that a randomly chosen piston fits in a randomly chosen cylinder is .97725, (correct to five significant figures).
  [8]
2. A random sample of ten such piston-cylinder assemblies is taken. Find the probability, (correct to two significant figures), that at least one assembly has a piston that does not fit in its cylinder.
  [7]

Four components are assembled into a system as shown.

A signal can flow through a subsystem connected in parallel (such as A-B) if at least one of the two components is working. A signal can flow through a subsystem connected in series only if both of its components are working. The probability that a component is working is P[A] = P[B] = .50 , P[C] = .80 and P[D] = .75 . The reliability of each component is independent of all of the other components. Find the probability that a signal can flow through the system from X to Y.

[15]

The heat output of 20 randomly selected motors performing certain tasks is measured. A new ventilation system is installed on those 20 motors and the heat output is measured again. An agent claims that the new ventilation system increases the heat output. It is known that both populations are normally distributed.
1. Which of the two sample t-tests (paired or unpaired) should be conducted?
  State the reason for your selection.
  [3]
2. Conduct the appropriate hypothesis test, at a level of significance of 1%, using whichever one of the two sets of Minitab^® output is appropriate:
```
Paired T for New - Old
             N    Mean  StDev  SE Mean
New         20  167.24  22.06     4.93
Old         20  157.18  24.67     5.52
Difference  20   10.06  10.80     2.41

99% lower bound for mean difference: 3.93
T-Test of mean difference = 0 (vs > 0): T-Value = 4.17  P-Value = 0.000

Two-sample T for New vs Old
      N   Mean    StDev   SE Mean
New  20  167.24   22.06      4.93
Old  20  157.18   24.67      5.52

Difference = mu (New) - mu (Old)
Estimate for difference:  10.06
99% lower bound for difference:  -7.91
T-Test of difference = 0 (vs >): T-Value = 1.36  P-Value = 0.091  DF = 38
Both use Pooled StDev = 23.4053
```
  [10]
3. Does the evidence support the agent’s claim?
  [2]

A random sample of fifty test objects is measured before (x) and after (y) a recalibration of the measuring device. A plot of the observed values of (x, y) and a normal probability plot of the residuals from simple linear regression are shown here.
1. State two reasons why a simple linear regression model for Y as a function of x is appropriate in this case. [2]
  
  Minitab^® produces the regression equation y = 1.05298 x – 22.722 and the following ANOVA table for these data.
```
    Source          DF      SS     MS       F      P
    Regression       1   11545  11545  309.40  0.000
    Residual Error  48    1791     37
    Total           49   13336
```
2. How much of the total variation in Y is explained by the simple linear regression model? [2]
3. Use the Minitab output above to conduct an appropriate hypothesis test to determine whether or not there is a significant linear association between Y and x . [4]
4. Use the information above, together with
  
  to construct a 95% prediction interval for a future observation of Y when x = 50 . [7]

A transmitter sends only two types of signal, 'A' and 'B'. In any one transmission, exactly one of these signals is sent, with signal 'A' being twice as likely as signal 'B'. After passage through a relay system, one of three signals, 'X', 'Y' or 'Z' is received. The following conditional probabilities are known:
1. Show that ,
  so that receipt of signal 'X' will result in the assumption that signal 'A' was sent.
  [4]
2. Find which of signals 'A', 'B' is more likely, if signal 'Y' is received.
  
  [4]
3. Find which of signals 'A', 'B' is more likely, if signal 'Z' is received.
  
  [3]
4. Hence find the probability that this system correctly determines which of signals 'A', 'B' was transmitted.
  
  [4]

BONUS QUESTION:

A random sample of 1000 people is taken, in order to determine whether or not a majority of the population would answer “yes” to a certain question.

Given that the true population proportion of the population that would answer “yes” to the question is 55%, find the probability (correct to two significant figures) that the random sample would lead to the appropriate hypothesis test (at a level of significance of a = 0.05) returning an incorrect conclusion.

[+6]

[Also provided with this examination paper were tables of the standard normal c.d.f. (the z tables)
and of the critical values of the t distribution (the t tables).]

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Created 2007 12 10 and most recently modified 2007 12 10 by Dr. G.H. George.