Portage Learning Math 110 Module 10 Exam
1 Find the value of X2 for 17 degrees of freedom and an area of .005 in the right
tail of the chi-square distribution.
Look across the top of the chi-square distribution table for .005, then look down the
left column for 17. These two meet at X2 =35.718.
2. Find the value of X2 for 10 degrees of freedom and an area of .005 in the left tail of the chi-
square distribution.
Since the chi-square distribution table gives the area in the right tail, we must use 1
- .005 = .995. Look across the top of the chi-square distribution table for .995, then
look down the left column for 10. These two meet at X2 =2.156.
3. Find the value of X2 values that separate the middle 80 % from the rest of the distribution for 9
degrees of freedom.
In this case, we have 1-.80=.20 outside of the middle or .20/2 = .1 in each of
the tails.
Notice that the area to the right of the first X2 is .80 + .10 = .90. So we use
this value and a DOF of 9 to get X2 = 4.168.
The area to the right of the second X2 is .10. So we use this value and a DOF of
9 to get X2 = 14.684.
, 4. Find the critical value of F for DOF=(4,17) and area in the right tail of .05
In order to solve this, we turn to the F distribution table that an area of .05.
DOF=(4,17) indicates that degrees of freedom for the numerator is 4 and degrees of
freedom for the denominator is 17. So, we look up these in the table and find that
F=2.96.
5. The mayor of a large city claims that 30 % of the families in the city earn more than $
100,000 per year; 52 % earn between $ 30,000 and $ 100,000 (inclusive); 18 % earn less than $
30,000 per year.
In order to test the mayor’s claim, 285 families from the city are surveyed
and it is found that:
90 of the families earn more than $ 100,000 per year;
135 of the families earn between $ 30,000 and $ 100,000 per
year (inclusive); 60 of the families earn less $ 30,000.
Test the mayor’s claim based on 5 % significance level.
We will set
H0: The mayor’s distribution is
correct. H1: The mayor’s
distribution is not correct.
This is a multinomial experiment, for multinomial experiments, we use the
chi-square distribution.
Calculate the degrees of freedom for three possible outcomes: DOF=3-1=2.
Our level of significance is 5% (.05). So look up DOF of 2 and .05 on the Chi-
square distribution table to get 5.991.
For the expected frequencies, we will use Ei = npi. So, the expected
frequencies (of the n=285 in the sample) based on the mayor’s distribution:
1 Find the value of X2 for 17 degrees of freedom and an area of .005 in the right
tail of the chi-square distribution.
Look across the top of the chi-square distribution table for .005, then look down the
left column for 17. These two meet at X2 =35.718.
2. Find the value of X2 for 10 degrees of freedom and an area of .005 in the left tail of the chi-
square distribution.
Since the chi-square distribution table gives the area in the right tail, we must use 1
- .005 = .995. Look across the top of the chi-square distribution table for .995, then
look down the left column for 10. These two meet at X2 =2.156.
3. Find the value of X2 values that separate the middle 80 % from the rest of the distribution for 9
degrees of freedom.
In this case, we have 1-.80=.20 outside of the middle or .20/2 = .1 in each of
the tails.
Notice that the area to the right of the first X2 is .80 + .10 = .90. So we use
this value and a DOF of 9 to get X2 = 4.168.
The area to the right of the second X2 is .10. So we use this value and a DOF of
9 to get X2 = 14.684.
, 4. Find the critical value of F for DOF=(4,17) and area in the right tail of .05
In order to solve this, we turn to the F distribution table that an area of .05.
DOF=(4,17) indicates that degrees of freedom for the numerator is 4 and degrees of
freedom for the denominator is 17. So, we look up these in the table and find that
F=2.96.
5. The mayor of a large city claims that 30 % of the families in the city earn more than $
100,000 per year; 52 % earn between $ 30,000 and $ 100,000 (inclusive); 18 % earn less than $
30,000 per year.
In order to test the mayor’s claim, 285 families from the city are surveyed
and it is found that:
90 of the families earn more than $ 100,000 per year;
135 of the families earn between $ 30,000 and $ 100,000 per
year (inclusive); 60 of the families earn less $ 30,000.
Test the mayor’s claim based on 5 % significance level.
We will set
H0: The mayor’s distribution is
correct. H1: The mayor’s
distribution is not correct.
This is a multinomial experiment, for multinomial experiments, we use the
chi-square distribution.
Calculate the degrees of freedom for three possible outcomes: DOF=3-1=2.
Our level of significance is 5% (.05). So look up DOF of 2 and .05 on the Chi-
square distribution table to get 5.991.
For the expected frequencies, we will use Ei = npi. So, the expected
frequencies (of the n=285 in the sample) based on the mayor’s distribution:
