MidTerm Stat 160	Spring Term 2003	Feb. 27 th	NAME:
	FORM A

1.

Customers arrive at a travel agency at a mean rate of 11 per hour. Assuming that the number of arrivals per hour has a Poisson distribution, determine the probability that more than 10 customers arrive in a given hour.

Rweb:> # CUMULATIVE POISSON DISTRIBUTION  
Rweb:> ppois(10, 11)  
[1] 0.4598887 

Rweb:> # POISSON PROBABILIY  
Rweb:> dpois(10, 11)  
[1] 0.1193781

(a).

0.5401

(b).

0.4598

(c).

0.1193

(d).

0.8714

2.

Consider the experiment of drawing a sample of chips, from a bucket. The bucket contains 20 numbered chips. Estimate the probability of seeing a number divisible by 2.

   Number of trials = 10
   Minimum value = 1
   Maximum value = 20
   Number to sample = 1
   With Replacement

Trial 1
13      
 
Trial 2
20      
 
Trial 3
18      
 
Trial 4
10      
 
Trial 5
7       
 
Trial 6
5       
 
Trial 7
7       
 
Trial 8
13      
 
Trial 9
19      
 
Trial 10
1

(a).

$\widehat{p}= 0.6$

(b).

$\widehat{p}= 0.7$

(c).

$\widehat{p}= 0.2$

(d).

$\widehat{p}= 0.3$

3.

What is the error of the estimated probability for the previous problem?.

(a).

0.042

(b).

0.145

(c).

0.289

(d).

0.458

4.

A stereo system contains 3 transistors. The probability that a given transistor will fail in 100,000 hours of use is .01. Assuming that the failures of the various transistors are independent of each other, what is the probability that at least one transistor will operate without failure for the first 100,000 hours?.

(a).

0.999999

(b).

0.000001

(c).

0.001

(d).

0.9925

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For the next three problems:
The following data represent the number of calories per half-cup serving of 15 brands of chocolate ice cream:

5.

What is the median calories per half-cup serving of 15 brands of chocolate ice cream?

(a)

160

(b)

150

(c)

170

(d)

190

6.

What is the value of the interquartile range of the data set?

(a)

40

(b)

80

(c)

20

(d)

110

7.

Identify any outliers in this data set?

(a)

No outliers.

(b)

110.

(c)

110 and 190

(d)

180

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8.

A psychologist is studying self-esteem and its effects on bulimic tendencies. The psychologist determined that the scores on a self-esteem test taken by patients diagnosed with bulimia are normally distributed with a mean of 55 and a standard deviation of 5. According to the empirical rule, what is the range of test scores for 99.5% of the bulimia patients?

(a.)

40-70

(b.)

5-55

(c.)

50-60

(d.)

45-65

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For the next four problems: The 1980 EPA 49-state (all except California) combined mileage rating and engine volume are listed in the following table for nine standard-transmission, four cylinder, gasoline-fueled, subcompact cars. The engine sizes are in total cubic inches of cylinder volume.

cylinder volume(x)	combined mpg(y)
97	24
85	29
98	26
105	24
120	24
151	22
140	23
134	23
146	21

Suppose we are interested in prediciting mpg in terms of cylinder volume. The least squares procedure gave $\widehat{Y} = 34.00 - 0.08X$ and R²=74.8

9.

Using the regression equation(if possible), predict combined mpg given a cylinder volume of 100.

(a)

34

(b)

66

(c)

74

(d)

26

10.

Which of the following is the best interpretation of the slope of the regression equation?

(a)

When the cylinder volume increases by 1, combined mpg increases by 0.08.

(b)

When the cylinder volume increase by 1, combined mpg decreases by 0.08.

(c)

When the cylinder volume is equal to zero, combined mpg is 34.

(d)

It has no practical meaning as cylinder volume equal to zero is not in the range of the data.

11.

Which of the following is the best interpretation of the coefficient of determination, R²?

(a)

When the cylinder volume is equal to zero, combined mpg increases by 74.8%.

(b)

For every 1 point increase in combined mpg, cylinder volume increases by 74.8.

(c)

74.8% of variation in combined mpg is accounted for by cylinder volume.

(d)

25.2% of variation in combined mpg is accounted for by cylinder volume.

12.

A compact car has a cylinder volume of 140 and a combined mpg of 23. Using the regression model, calculate the residual for this car.

(a)

22.8

(b)

0.20

(c)

0.08

(d)

25.2

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13.

Suppose you are interested in predicting the future wealth of a young male. You have concluded that the number of sisters he has won't help in predicting his future wealth. Hence, you think that the events ``Future wealth for young males'' and ``Number of sisters he has'' are

(a)

Disjoint events.

(b)

Dependent events.

(c)

Independent events.

(d)

Events of probability 0.

14.

A box contains 5 good bulbs and 2 defective ones. If two bulbs are selected (without replacement), what is the probability of both the bulbs being defective?. Use a tree diagram to get the answer.

(a).

0.476

(b).

0.046

(c).

0.081

(d).

0.510

15.

An experiment consists of selecting a number at random from the set of numbers 1,2,3,4,5,6,7,8,9. Find the probability that the number selected is less than 4.

(a).

0.56

(b).

0.67

(c).

0.33

(d).

0.44

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The next four questions pertain to the following information:

The commuting time to campus for a typical WMU student is normally distributed with a mean of 27 minutes and standard deviation of 8 minutes. Use the class code results from RWEB to answer the questions given below. For example, letting X denote the student's commuting time P(X<30) is displayed as:

Rweb:> # CUMULATIVE NORMAL DISTRIBUTION  
Rweb:> pnorm(30, 27, 8)  
[1] 0.6461698

If the value of k is desired such that P(X<k)= .75, then the result is displayed as:

Rweb:> # NORMAL PERCENTAGE POINT  
Rweb:> qnorm(.75, 27, 8)  
[1] 32.39592 



RWEB Results:

Rweb:> # NORMAL PERCENTAGE POINT  
Rweb:> qnorm(.95, 27, 8)  
[1] 40.15883 

Rweb:> # CUMULATIVE NORMAL DISTRIBUTION  
Rweb:> pnorm(30, 27, 8)  
[1] 0.6461698 
Rweb:> # CUMULATIVE NORMAL DISTRIBUTION  
Rweb:> pnorm(15, 27, 8)  
[1] 0.0668072
Rweb:> # NORMAL PERCENTAGE POINT  
Rweb:> qnorm(.05, 27, 8)  
[1] 13.84117 
Rweb:> # NORMAL PERCENTAGE POINT  
Rweb:> qnorm(.15, 27, 8)  
[1] 18.70853 
Rweb:> # CUMULATIVE NORMAL DISTRIBUTION  
Rweb:> pnorm(.15, 27, 8)  
[1] 0.0003950356 
Rweb:> # CUMULATIVE NORMAL DISTRIBUTION  
Rweb:> pnorm(30, 27, .8)  
[1] 0.9999116 
Rweb:> # CUMULATIVE NORMAL DISTRIBUTION  
Rweb:> pnorm(27, 30, .8)  
[1] 0.00008841729

16.

What is the probability that the commuting time for a student exceeds 15 minutes?

(a.)

.9331928

(b.)

18.70853

(c.)

.0668072

(d.)

.0003950356

17.

95% of the students will arrive in less than what time?

(a.)

27

(b.)

13.84117

(c.)

.95

(d.)

40.15883

18.

What is the probability that the commuting time for a student will take between 15 and 30 minutes?

(a.)

.0668072

(b.)

.6461698

(c.)

.5793626

(d.)

.0003950356

19.

If we took a random sample of 100 WMU students, what is the probability that the average arrival time for these 100 students would be less than 30 minutes?

(a.)

.00008841729

(b.)

.6461698

(c.)

.5793626

(d.)

.999916

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20.

When estimating parameters, under which of the following conditions would we expect the mean and median of a sample to be equal (or very close) to each other?

(a.)

When the distribution is skewed

(b.)

When the distribution is symmetric

(c.)

When the standard deviation is less than 0

(d.)

When the standard deviation is greater than 0

21.

Which of the following is TRUE about the sample variance?

(a)

50% of the data is less than the sample variance.

(b)

The sample variance is not affected by outliers.

(c)

The sample variance is the average squared deviations from the mean.

(d)

The sample variance is one of the 5 basic statistic.

22.

Which of the following is a robust statistic?

(a)

Hodges-Lehman

(b)

Mean

(c)

Standard deviation

(d)

Range

23.

Roll 5 dice at the same time, where each die is balanaced and has the numbers 1 through 6 on it. If your total exceeds 18 you win. You decide to perform 100 resampling trials. Which of the following resampling models would be the correct model to simulate the probability of winning.

(a).

   Number of trials = 100
   Minimum value = 1
   Maximum value = 6
   Number to sample = 5
   With Replacement

(b).

   Number of trials = 10
   Minimum value = 1
   Maximum value = 6
   Number to sample = 5
   With Replacement

(c).

   Number of trials = 100
   Minimum value = 1
   Maximum value = 6
   Number to sample = 5
   Without Replacement

(d).

   Number of trials = 100
   Minimum value = 1
   Maximum value = 6
   Number to sample = 1
   With Replacement

24.

Suppose we have a data on quarterly sales tax receipts (in thousand dollars) which is left-skewed with a very large outliers on the left side. Which of the following statement will be generally correct?

(a)

The sample mean quarterly sales tax receipts would lie to the left of the sample median quarterly sales tax receipts.

(b)

The sample mean quarterly sales tax receipts would lie to the right of the sample median quarterly sales tax receipts.

(c)

The sample mean quarterly sales tax receipts and sample median quarterly sales tax receipts will be the same.

(d)

The sample median quarterly sales tax receipts would lie to the left of the sample mean quarterly sales tax receipts.

25.

A parachutist must land in the circle shown below. The target area surrounding this circle is a large field measuring 100 yards by 100 yards (total area is 10,000 square yards). The circle is 20 yards in diameter (total area is 314 square yards). Let X be a random variable representing the point where the parachutist lands. Assume that X follows a uniform distribution over the target area. Given this information, what is the probability that the parachutist will land in the circle?

 

Figure 0.1: Parachute Target Area

(a.)

0.0314

(b.)

0.314

(c.)

0.9686

(d.)

0.0686