Math 252 Final Exam Questions:                                    Name and ID Number:

 

 

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7

EXPACTATÝON(5P)

 

 

 

 

 

 

 

 

 

NOT: SHOW ALL YOUR WORK. ANSWERS WITHOUT WORK WILL NOT BE GRADED!!!

1. [30 points] The monthly rental for a two-bed room apartment in  Istanbul is 500 YTL. Assume that the population mean is 500YTL and the population standard deviation is 100YTL.

a) What is the probability that a simple random sample of a 49 two bedroom apartments will provide a sample mean monthly rental within75YTL of the population mean?

                b) What is the probability that a simple random sample of a 49 two bedroom apartments will provide a sample mean monthly rental within25YTL of the population mean?

c) Discuss the results in parts (a) and (b). Compare the results. In order to have same probability in part (b) what should we do?

 

 

Solutions:

a)        P(-75<<+75)=P()

P(-75<<+75)=P()                     

P(-75<<+75)=P(-5,25<Z<5,25)=almost 1

                b)

P(-25<<+25)=P()

P(-25<<+25)=P()                     

P(-25<<+25)=P(-1,75<X<1,75)=0,92

c) In the second case probability is less than the first one because margin of error around the mean is smaller in the second case. In order to have same probability in the second case we need to increase the sampling size.

 

2. [20 points] The number of hours Turks sleep each night varies considerably with 12% of the population sleep less than six hours to 3% sleeping more than eight hours. The following sample of 25 individuals reports the hours of sleep per night

6,5  7,3  5,6  5,1  6,5   6,6  6,9  6,2  6,4  7,1  7,9   8,5  6,0   6,9   6,6  7,5  7,2  8,6  7,1  5,5   6,4  6,7  7,7  5,9  6,3

a) What is the point estimate of the population mean number of hours of sleep each night?

b) Assume that the population has a normal distribution; develop a 95% confidence Interval for the population mean number of hours of sleep each night

                Solution:

                a)

b)

 

 

 

3. [20 points] A Turkish Retail Survey found that 9% of Turkish customers used internet to buy gifts during the 2005 holiday season. If 1089 customers participated in the survey, what is the margin of error and what is the interval estimate of the population proportion of customers using the internet to buy gifts? Use a 95% confidence.

4. [30 points] Suppose a new production method will be implemented if a hypothesis test supports the conclusion that a new method reduces the mean operating cost per hour.

a) State the appropriate null and alternative hypothesis if the mean cost for the current production method is 120YTL

b) What is the TYPE 1 error in this situation? What are the consequences of making this error?

c) What is the TYPE 2 error in this situation? What are the consequences of making this error?

 

a)

 

        Null Hypothesis: >120 Alternative: <120

c)        Claiming <120YTL when the new method does not lower costs. A mistake could be implimenting the method when it does not help

d)       Concluding >120YTL when the method really would lower costs. This could lead to not implementing a method that would lower the costs.

 

5. [35 points] The population mean earnings per share for financial services corporation including AK BANK, GARANTÝ, FORTIS, ÝÞ BANKASI and CITI BANK was 3YTL. A sample of 11 financial services corporation provided the followings per share data:

1,95 2,66         3,79         4,01         3,17         3,05         2,39         2,2           3,14         2,98         2,66

a) Formulate the null and alternative hypotheses that can be used to determine whether the population means earnings per share in 2001 differ from 3YTL reported in 2000

b) Using =0.05, what are critical values for the statistic, and what is the rejection rule

c) Compute the sample mean

d) Compute the sample standard deviation

e) Compute the value of test statistic.

f) What is your conclusion?

g) Approximate the P-Value.

 

Solution

a)

b)

Then The rejection Rule is t>2,228 or t<-2,228

c)

       

d)

e)

 

 

f)        since -0,1435>-2,228 then we retain the null hypothesis

g)        P-Value=2P(t<-0,1435)=2P(t>0,1435)=0,888746 (EXACT)

APPROXIMATION                                                                          

6. [35 points] The following shell of an ANOVA source table needs to be completed then answer a through

a)       To what does SOURCE refer?

b)       How many group means are being compared? How do you know this?

c)       How many total subjects were in the study? How do you know this?

d)       How many subjects are in each group? How do you know this?

e)       How many degrees of freedom are there?

f)        Was the null hypothesis rejected or retained? WHY?

g)       Two of the group means are 64.2 and 31.7. Can you use these mean values to interpret this finding for independent variable X and dependent variable Y? WHY?

Solution:

a)        Source is telling us where the total variabilities around the overall mean is caming from.

b)       Since MS=156/(k-1) then k-1=156/52 then k-1=3 and k=4 so there are 4 groups

c)         

d)       There is not sufficent information to find number of elements of each group

e)        3 for Between Groups and 19 for Within groups

f)        Since F=3,64 >3,133 then we can reject the Null Hypotesis

g)        No because there is not any relation between the means of the groups since we reject the Null Hypotesis

7. [30 points] With the growth of internet service providers, a researcher decides to examine whether there is a correlation between cost of internet service per month (rounded to the nearest dollar) and degree of customer satisfaction (on a scale of 1 - 10 with a 1 being not at all satisfied and a 10 being extremely satisfied). The researcher only includes programs with comparable types of services. A sample of the data is provided below.

dollars

satisfaction

11

6

15

4

9

9

5

6

19

5

25

10

 

a)        Develop a scatter diagram for these data with the dollar is independent variable and satisfaction dependent variable

b)       Develop an estimated regression equation that can be used to predict satisfaction given the amount of dollar spent for internet users

c)        Use the estimated regression equation to predict satisfaction score with 20 dollars spending for the service.

Solution:

a)

               

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

               

 

 

 

               

b)        

 

X

Y

x-meanx

y-meany

(x-meanx)*(y-meany)

(x-meanx)^2

 

11

6

-3

-0,66667

2

9

 

15

4

1

-2,66667

-2,666666667

1

 

9

9

-5

2,333333

-11,66666667

25

 

5

6

-9

-0,66667

6

81

 

19

5

5

-1,66667

-8,333333333

25

 

25

10

11

3,333333

36,66666667

121

 

 

 

 

 

 

 

TOTAL

84

40

 

 

22

262

 

14

6,666667

 

b1=

0,083969466

 

 

 

 

 

b0=

5,491094148

 

 

                               y=5,49+0,08X

 

c)                y=5,49+0,08*20

                                y=5,49+0,16=5,65 close to 6