Probability & Statistics

19. Suppose a sample of small earthquakes  magnitude 6.0 occurs on days of the week with
these frequencies:
Sun Mon Tue Wed Thu Fri Sat
6 7 5 8 5 12 6
Looking at the data, Darcy wonders if quakes happen more on Fridays.
a. What is the P-value for a z-test of proportion with 1a Friday 7H : ?p 
The sample proportion is
sFriday
12
0.245
49
p  
while the sample standard deviation is
 
  
s s
Friday Friday1
1
0.245 0.755
48
0.062.
p p
s
n





Thus, the z-value of the observed Friday proportion of earthquakes is given by
s
Friday 0
0.245 0.143
0.062
1.65.
p
z
s





The P-value corresponding to this z-value is given by
0.100.P 
b. We wish to determine the P-value for a 2 goodness of fit test with 0H : Quakes are
evenly distributed throughout the week.
To do so, we first enlarge the above table to include the expected number of earthquakes
observed on each day of the week.
Day Sun Mon Tue Wed Thu Fri Sat Total
Observed number 6 7 5 8 5 12 6 49
Expected number 7 7 7 7 7 7 7 49
Thus we have
 
             
2
7
2
1
2 2 2 2 2 2 2
6 7 7 7 5 7 8 7 5 7 12 7 6 7
7 7 7 7 7 7 7
1 0 4 1 4 25 1
7
36
7
5.14.
i i
i i
O E
E




      
      
     




The corresponding P-value with 6 degrees of freedom is 0.526.
20. Chase asks 187 students whether they consider themselves to be “left”, “center”, or “right”
politically, and whether they believe in God, in some other type of universal spirit, or neither.
The data is tabulated below:
God Spirit Neither Total
Left 30 38 16 84
Center 30 24 10 64
Right 24 12 3 39
Total 84 74 29 187
a. Find the smallest expected value using the formula

  
 
row total column total
expected count .
grand total

The following table shows the expected counts for each category:
God Spirit Neither Total
Left 37.7 33.2 13.0 84
Center 28.7 25.3 9.9 64
Right 17.5 15.4 6.1 39
Total 84 74 29 187
As can be seen from the above table, each entry is at least 5.
b. We wish to write hypotheses for a 2 test of independence, using a level of significance
of 0.1  and 4 degrees of freedom. [1]

Our null hypothesis is as follows: 20H : 7.78. 
Our alternate hypothesis is as follows: 2aH : 7.78. 
c. We wish to carry out the test.

We have
 
     
     
     
2
3
, ,2
, 1 ,
2 2 2
2 2 2
2 2 2
30 37.7 38 33.2 16 13.0
37.7 33.2 13.0
30 28.7 24 25.3 10 9.9
28.7 25.3 9.9
24 17.5 12 15.4 3 6.1
17.5 15.4 6.1
7.83.
i j i j
i j i j
O E
E




  
  
  
  
  
  


Thus we would go with the alternative hypothesis at the 10% level of significance. The P-
value for 2 here is 0.098.
d. As can be seen from the above tables, those who label themselves as “right” tend to
believe in God more and believe in neither God nor a supreme spirit less than those who
label themselves as “left” or “center”. The other differences are less significant.
21. A sample of online courses is compared to a sample of face-to-face courses at Lane
Community College. Counts of grades awarded in each sample are shown in this table along with
relative frequencies.
counts relative frequencies
online face-to-face online face-to-face
A 3,257 39,838 0.44 0.49
B 2,149 22,523 0.29 0.28
C 1,175 11,584 0.16 0.14
D 327 3,049 0.04 0.04
F 466 3,554 0.06 0.04
total 7,374 80,548 1 1
a. We wish to write hypotheses for a 2 test  4 degrees of freedom, 0.05 

2
0H : 9.49. 
2
aH : 9.49. 
b. We wish to carry out the test and find P.

We have the following expected values:
expected counts
online face-to-face
A 3,614 39,481
B 2,069 22,603
C 1,070 11,689
D 283 3,093
F 337 3,683
total 7,374 80,548
Thus we have
 
   
   
   
   
   
2
5 2
, ,2
1 1 ,
2 2
2 2
2 2
2 2
2 2
3,257 3,614 39,838 39,481
3,614 39,481
2,149 2,069 22,523 22,603
2,069 22,603
1,175 1,070 11,504 11,689
1,070 11,689
327 283 3,049 3,093
283 3,093
466 337 3,554 3,683
337 3
i j i j
i j i j
O E
E

 


 
 
 
 
 
 
 
 
 
 

,683
114.48.
The corresponding P-value is essentially zero. Thus, this test shows conclusively that
there is a significant difference between the grades of online and face-to-face students.
c. I would describe this as a high-power test since the...