252solngr1 9/16/05 (Open this document in 'Page Layout' view!)
Graded Assignment 1
Please show your work! Neatness and whether the papers are stapled may affect your grade.
1. A magazine wants to estimate the mean leisure time in hours enjoyed weekly by managers. Data taken from a sample of 21 managers follows:
15 12 18 23 11 21 16 13 9 19 26 11 7 18 11 15 23 26 10 8 17
Compute the sample standard deviation using the computational formula. Use this sample standard deviation to compute a 98% confidence interval for the mean. Does the mean differ significantly from 19 hours? Why?
2. How would your results change if the sample of 21 had been taken from a population of 100?
3. Assume that the population standard deviation is 6.00 (and that the sample of 21 is taken from a very large population). Find using the Normal table (If you have several values of that you can use, pick the average of the extreme ones.) and use it to compute a 99.9% confidence interval. Does the mean differ significantly from 19 hours? Why?
Solution:
1)
252solngr1 9/16/05 (Open this document in 'Page Layout' view!)
index
1 15 225
2 12 144
3 18 324
4 23 529
5 11 121
6 21 441
7 16 256
8 13 169
9 9 81
10 19 361
11 26 676
12 11 121
13 7 49
14 18 324
15 11 121
16 15 225
17 23 529
18 26 676
19 10 100
20 8 64
21 17 289
sum 329 5825
,,
The formula for the sapmle standard deviation is in Table 20 of the Supplement.
.
252solngr1 9/16/05 (Open this document in 'Page Layout' view!)
From Table 3 is the formula for a two sided confidence interval when the population standard deviation is unknown. or 12.472 to 18.861. If we ask if the mean is significantly different from 19, our null hypothesis is and since 19 isnot between the top and the bottom of the confidence interval,reject and say that the mean is significantly different from 19. (But it is not significantly different from 18!)
2) If , the sample of 15 is more than 5% of the population, so use .
Recall that , , and .
Confidence interval: is the formula for a two sided interval. or 13.118 to 18.215. The interval is smaller, but it doesn’t change anything – the mean is still significantly different from 19(but not 18).
3) a) Find .
Make a diagram! The diagram for will be a Normal curve centered at zero and will show one point, , which has 0.05% above it (and 99.95% below it!) and is above zero because zero has 50% below it. Sincezero has 50% above it, the diagram will show 49.95% between zero and .
From the diagram, we want one point so that or . On the interior of the Normal table we can find to .4995 exactly. In fact, it says for any value of between 3.27 and 3.32. A compromise seems to be required, so note that the halfway point between these numbers is 3.295. This means that we will say.
Check: This is verified graphically below.
b) We know that , and . So
=1.3093. The 99.9% confidence interval has or , so . The confidence interval is or 10.53 to20.81. If we test the null hypothesis against the alternative hypothesis ,since 19 is on the confidence interval, we do not reject the null hypothesis.
Extra Credit:
4) a. Use the data above to compute a 98% confidence interval for the population standard deviation.
Solution: From the supplement pg 1, . We know , ,,and . We use and . The formula becomes or . If we take square roots, we get .
b. Assume that you got the sample standard deviation that you got above from a sample of 45, repeat a.
Solution: From the supplement pg 2, . We now have , , and . We use and . The formula becomes or .
c. Fool around with the method for getting a confidence interval for a median and try to come close to a 99% confidence interval for the median.
The numbers in order are
7 8 9 10 11 11 11 12 13 15 15 16 17 18 18 19 21 23 23 26 26
It says on the outline that . We want to be 1% or lower which means . Unfortunately, our Binomial table only goes to and , and if we look where , we find that the first quantity at or below 0.5% is for and for . So we can only guess that is between 5 and 6. To be more accurate, use the formula .It tells us to use the 5th number from the end, since, if we want to be conservative we round the answer down.
I can check my results two ways. I am using a continuity correction, which adds 0.5 to each interval.
I also checked it by generating a binomial table for and on Minitab. I put the numbers 0 through 7 in C1, used the Calcmenu, then Probability Distributions and then Binomial. In the dialog box I picked, Cumulative probability, Number of trials= 21 , Probability of success =.5, Input column = C1 and Optional storage = C2. The equivalent command is:
MTB > CDF c1 c2;
SUBC> Binomial 21 0.5.
The results, in C1 and C2 are:
00.0000005
10.0000105
20.0001106
30.0007448
40.0035987
50.0133018
60.0391769
70.0946236
How I got these results
‘MTB >’ is the Minitab prompt. The retrieval is done using the ‘file’ pull-down menu and the ‘open worksheet’ command followed by finding where I put the data. Other instructions were typed in the ‘session’ window.
I put the data in column 1 in Minitab and used the ‘Gsummary’ and ‘Describe’ commands to get the mean and standard deviation.
Session to get confidence intervals
————— 9/16/2005 5:34:34 PM ————————————————————
Welcome to Minitab, press F1 for help.
Results for: 2gr1-052.MTW
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\2gr1-052.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\2gr1-052.MTW'
Worksheet was saved on Wed Sep 07 2005
MTB > Describe c1
Descriptive Statistics: time
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum
time 21 0 15.67 1.26 5.79 7.00 11.00 15.00 20.00 26.00
MTB > Gsummary c1;
SUBC> confidence 98.
Summary for time
MTB > let c2 = c1*c1I computed the square of C1 in C2..
MTB > sum c1I got the sums for computing the variance.
Sum of time
Sum of time = 329
MTB > sum c2
Sum of C2
Sum of C2 = 5825
MTB > print c1 c2
Data Display
Row time C2
1 15 225
2 12 144
3 18 324
4 23 529
5 11 121
6 21 441
7 16 256
8 13 169
9 9 81
10 19 361
11 26 676
12 11 121
13 7 49
14 18 324
15 11 121
16 15 225
17 23 529
18 26 676
19 10 100
20 8 64
21 17 289
MTB > Onet c1;This does a 98% confidence interval and a test for a mean of 19 using s.
SUBC> test 19;
SUBC> confidence 98.
One-Sample T: time
Test of mu = 19 vs not = 19
Variable N Mean StDev SE Mean 98% CI T P
time 21 15.6667 5.7908 1.2637 (12.4722, 18.8612) -2.64 0.016
MTB > Onez c1;This does a 99.9% confidence interval and a test for a mean of 19 using sigma.
SUBC> sigma 6;
SUBC> confidence 99.9.
One-Sample Z: time
The assumed standard deviation = 6
Variable N Mean StDev SE Mean 99.9% CI
time 21 15.6667 5.7908 1.3093 (11.3584, 19.9750)
MTB > let c5=c1
MTB > sort c5 c5;This sorts c1, which I moved to c5
SUBC> by c5.
MTB > print c5
Data Display
times
7 8 9 10 11 11 11 12 13 15 15 16 17 18 18
19 21 23 23 26 26
Session to verify value of z
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\notmuch.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\notmuch.MTW'This worksheet is nonsense and not actually used..
Worksheet was saved on Thu Apr 14 2005 It gets Minitab to look in the same place for normarea.
Results for: notmuch.MTW
MTB > %normarea5aThis does the graph shown below. It prompts you to put in the data.
Executing from file: normarea5a.MAC
Graphic display of normal curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K116)
Enter the mean and standard deviation of the normal curve.
DATA> 0
DATA> 1
Do you want the area to the left of a value? (Y or N)
no
Do you want the area to the right of a value? (Y or N)
yes
Enter the value for which you want the area to the right.
DATA> 3.295
...working...
Normal Curve Area
Extra Credit Minitab Problem
5. Check some numbers in the t, Chi-Squared or F tables using the new set of Minitab routines that I have prepared. To use the new set of routines, set up a file to hold your work. Then go to http://courses.wcupa.edu/rbove , open the Minitab folder and download any of the following:
Normal Distribution Area programs:
NormArea5A.txt
or NormArea5C.txt and NormArea5.txt.
t Distribution Area programs:
tAreaA.txt
or tAreaC.txt and tArea.txt
Chi-squared Distribution Area programs:
ChiAreaA.txt
or ChiAreaC.txt and ChiArea.txt
F Distribution area programs:
FAreaA.txt
or FAreaC.txt and FArea.txt
Use Notepad (under ‘tools’ in Minitab’) to convert their extensions from .txt back to .mac. To see how they are used, look at http://courses.wcupa.edu/rbove/Minitab/Area.doc.
Routines like tAreaA are self prompting. To use routines like tAreaC, you need to set up your data in advance. If you want to use one of the worksheets that are mentioned in http://courses.wcupa.edu/rbove/Minitab/Area.doc, click on ‘File’ and then ‘Open Worksheet.’ Copy a URL like the ones below into FileName.’
http://courses.wcupa.edu/rbove/Minitab/252PrA1d-f.MTW
http://courses.wcupa.edu/rbove/Minitab/tEx1.MTW
http://courses.wcupa.edu/rbove/Minitab/ChiEx1.MTW
http://courses.wcupa.edu/rbove/Minitab/FEx1.MTW
Addendum: To get graphs into a document do the following. While you are in Minitab, click on the graph and use the ‘File’ menu; choose ‘Save Graph as….’ A menu will appear. Pick a type (like ‘.jpg’ for color or ‘.png’ for black and white) under ‘Save as type …’ give the graph a name like ‘graph1.jpg’ and the graph will be placed in the same file that contains your data and macro. You can now use the ‘Insert’ menu in Word to insert a picture.
Results: I looked at the tables and found , , , , and. For the numbers with .10 as a subscript, I checked that the probability above them was .10, for the numbers with .90 as a subscript, I checked that the probability below them was .10.
————— 9/19/2005 5:33:43 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\notmuch.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\notmuch.MTW'
Worksheet was saved on Thu Apr 14 2005
Results for: notmuch.MTW
MTB > %tarea6a
Executing from file: tarea6a.MAC
Graphic display of t curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.
DATA> 10
Do you want the area to the left of a value? (Y or N)
n
Do you want the area to the right of a value? (Y or N)
y
Enter the value for which you want the area to the right.
DATA> 1.372
...working...
t Curve Area
Data Display
mode 0
median 0
MTB > %normarea6a
Executing from file: normarea6a.MAC
Graphic display of normal curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K116)
Enter the mean and standard deviation of the normal curve.
DATA> 0
DATA> 1
Do you want the area to the left of a value? (Y or N)
n
Do you want the area to the right of a value? (Y or N)
y
Enter the value for which you want the area to the right.
DATA> 1.282
...working...
Normal Curve Area
MTB > %chiarea6a
Executing from file: chiarea6a.MAC
Graphic display of chi square curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.
DATA> 10
Do you want the area to the left of a value? (Y or N)
n
Do you want the area to the right of a value? (Y or N)
y
Enter the value for which you want the area to the right.
DATA> 15.9872
...working...
ChiSquare Curve Area
Data Display
std_dev 4.47214
mode 8.00000
median 9.33333
MTB > %chiarea6a
Executing from file: chiarea6a.MAC
Graphic display of chi square curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.
DATA> 10
Do you want the area to the left of a value? (Y or N)
l
Please answer Yes or No.
y
Enter the value for which you want the area to the left.
DATA> 4.8650
...working...
Chi Squared Curve Area
Data Display
std_dev 4.47214
mode 8.00000
median 9.33333
MTB > %farea6a
Executing from file: farea6a.MAC
Graphic display of F curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.DF2 must be above 4.
DATA> 10
DATA> 10
Do you want the area to the left of a value? (Y or N)
n
Do you want the area to the right of a value? (Y or N)
y
Enter the value for which you want the area to the right.
DATA> 2.32
...working...
F Curve Area
Data Display
mode 0.818182
std dev 0.968246
MTB > %farea6a
Executing from file: farea6a.MAC
Graphic display of F curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.DF2 must be above 4.
DATA> 10
DATA> 10
Do you want the area to the left of a value? (Y or N)
y
Enter the value for which you want the area to the left.
DATA> .431
...working...
F Curve Area
Data Display
mode 0.818182
std dev 0.968246
MTB >