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MATH1005 Quizzes

Quiz 12: Chi-square tests
Question 1 Questions
If P(χ52 > a) = 0.1, find a. Exactly one option must be correct)
a)
a = 9.236
b)
a = 1.610
c)
a = 7.779
d)
a = 15.086

Choice (a) is correct!
Use the chi-square tables with ν = 5, and p = .1.
Choice (b) is incorrect
P(χ52 > 1.610) = 0.9.
Choice (c) is incorrect
You have used the wrong value for ν.
Choice (d) is incorrect
P(χ52 > 15.086) = 0.01.
Provide bounds for p = P(χ252 > 38.5). Exactly one option must be correct)
a)
0 p 0.01
b)
0.05 p 0.025
c)
0.025 p 0.05
d)
0.05 p 0.1

Choice (a) is incorrect
Try again.
Choice (b) is incorrect
You have reversed the upper and lower limits. Obviously P(χ252 > 38.5) must exceed 0.025.
Choice (c) is correct!
Draw a chi-square diagram and mark in the value 38.5 on the x-axis. From the tables, P(χ252 > 40.646) = 0.025 and P(χ252 > 37.652) = 0.05. Mark both 37.652 and 40.646 on the x-axis as well, and it will be obvious that P(χ252 > 38.5) has bounds (0.025, 0.05).
Choice (d) is incorrect
Draw a chi-square diagram and mark in the value 38.5 on the x-axis. From the tables, P(χ252 > 37.652) = 0.05. Now mark 37.652 on your diagram, and it will be obvious that P(χ252 > 38.5) must be less than 0.05.
Questions 3 to 6 use the same information.
The observed frequencies of genotypes A, B and C of 100 progeny from a genetic cross are Oi: 18, 55 and 27 respectively. A model states that A, B and C occur in the ratio 1:2:1.
Under the model, the expected frequencies (Ei) are: Exactly one option must be correct)
a)
Ei: 10, 20, 10
b)
Ei: 25, 50, 25
c)
Ei: 25, 25, 50
d)
Ei: 50, 25, 25

Choice (a) is incorrect
The frequencies must satisfy iEi = iOi.
Choice (b) is correct!
Under the model, the proportions for A, B and C are respectively 1 4,1 2,1 4. Multiplying by 100 gives the expected frequencies 25, 50, 25.
Choice (c) is incorrect
These are not in the correct order.
Choice (d) is incorrect
These are not in the correct order.
Questions 3 to 6 use the same information.
The observed frequencies of genotypes A, B and C of 100 progeny from a genetic cross are Oi: 18, 55 and 27 respectively. A model states that A, B and C occur in the ratio 1:2:1.
Writing Ei for the corresponding expected frequencies under the model, calculate τobs = i(OiEi)2 Ei , the observed value of the chi-square test statistic. Exactly one option must be correct)
a)
102.62
b)
2.62
c)
0.262
d)
0.78

Choice (a) is incorrect
You have found iOi2 Ei .
Choice (b) is correct!
τobs = (1825)2 25 + (5550)2 50 + (2725)2 25 = 2.62.
Choice (c) is incorrect
Check your calculations.
Choice (d) is incorrect
τobs = i(OiEi)2 Ei . You have found instead, i(OiEi)2 iEi .
Questions 3 to 6 use the same information.
The observed frequencies of genotypes A, B and C of 100 progeny from a genetic cross are Oi: 18, 55 and 27 respectively. A model states that A, B and C occur in the ratio 1:2:1.
State the appropriate chi-square variable, χν2 , for testing the goodness of fit of this model. Exactly one option must be correct)
a)
χ22
b)
χ12
c)
χ32
d)
χ992

Choice (a) is correct!
There are three categories, so ν = 2.
Choice (b) is incorrect
Try again. There are three categories
Choice (c) is incorrect
ν = k 1 where k is the number of categories.
Choice (d) is incorrect
There are 3 categories, not 100.
Questions 3 to 6 use the same information.
The observed frequencies of genotypes A, B and C of 100 progeny from a genetic cross are Oi: 18, 55 and 27 respectively. A model states that A, B and C occur in the ratio 1:2:1.
Test the model for goodness of fit, providing an expression for the P-value. Exactly one option must be correct)
a)
P-value = P(χ992 2.62) > 0.99. The model is a good fit.
b)
P-value = P(χ12 2.62) < 0.1. The data provide evidence against the model.
c)
P-value = P(χ22 2.62) is very large. The data are consistent with the model.
d)
P-value = P(χ22 2.62) > 0.1. The data are consistent with the model.

Choice (a) is incorrect
You have used the wrong chi-square distribution.
Choice (b) is incorrect
You have used the wrong chi-square distribution.
Choice (c) is incorrect
The P-value is not the lower tail of χ22.
Choice (d) is correct!
Since there are 3 categories and since τobs = 2.62, the appropriate chi-square distribution is χ22 and P-value = P(χ22 2.62) > 0.1. This means that the data are fairly typical of what is expected under the model.
Questions 7 to 10 use the same information.
A market researcher wishes to assess consumers’ preference among four different colours available on a name-brand dishwasher. In a sample of 198 recent sales, 61 were for stainless steel, 55 for white, 41 for black and 41 for cream.
To test the hypothesis that all four colours are equally preferred, a chi-square test is used. Writing Ei for the corresponding expected frequencies under the model, calculate τobs = i(OiEi)2 Ei , the observed value of the chi-square test statistic. Exactly one option must be correct)
a)
τobs = 6.2
b)
τobs = 10.3
c)
τobs = 206.7
d)
τobs = 204.2

Choice (a) is correct!
τobs = (6149.5)2 49.5 + (5549.5)2 49.5 + (4149.5)2 49.5 + (4149.5)2 49.5 = 6.2.
Choice (b) is incorrect
Check your calculations.
Choice (c) is incorrect
Check your calculations.
Choice (d) is incorrect
You have found iOi2 Ei instead of i(OiEi)2 Ei
Questions 7 to 10 use the same information.
A market researcher wishes to assess consumers’ preference among four different colours available on a name-brand dishwasher. In a sample of 198 recent sales, 61 were for stainless steel, 55 for white, 41 for black and 41 for cream.
To test the hypothesis that all four colours are equally preferred, a chi-square test is used. Write an expression for the P-value. Exactly one option must be correct)
a)
P-value = P(χ22 6.2) > 0.05
b)
P-value = P(χ32 6.2) > 0.1
c)
P-value = P(χ32 10.3) < 0.025.
d)
P-value = P(χ1972 6.2) > 0.99

Choice (a) is incorrect
You have used the wrong chi-square distribution.
Choice (b) is correct!
Under this model, the Ei are all 49.5, and τobs = 6.2. Since there are 4 categories, the appropriate chi-square is χ32, and P-value = P(χ32 6.2) > 0.1
Choice (c) is incorrect
Check your calculation of τobs.
Choice (d) is incorrect
Try again. You have used the wrong chi-square distribution.
Questions 7 to 10 use the same information.
A market researcher wishes to assess consumers’ preference among four different colours available on a name-brand dishwasher. In a sample of 198 recent sales, 61 were for stainless steel, 55 for white, 41 for black and 41 for cream.
Writing Ei for the expected frequencies under a model which states that the preferences are in the ratio 6:5:4:3, calculate τobs = i(OiEi)2 Ei , the observed value of the chi-square test statistic. Exactly one option must be correct)
a)
τobs = 200.52
b)
τobs = 8
c)
τobs = 0.495
d)
τobs = 2.52

Choice (a) is incorrect
You have found iOi2 Ei instead of i(OiEi)2 Ei
Choice (b) is incorrect
The expected frequencies under this model are 66, 55, 44, 33. Using these, recalculate τobs.
Choice (c) is incorrect
τobs = i(OiEi)2 Ei . You have found instead, i(OiEi)2 iEi .
Choice (d) is correct!
The expected frequencies under this model are 66, 55, 44, 33 (for example the model indicates that the proportion preferring stainless steel is 6 6+5+4+3 = 1 3 Multiplying by 198 gives 66. similarly for the other three colours.
Thus, τobs = (6166)2 66 + (5555)2 55 + (4144)2 44 + (4133)2 33 = 2.52.
Questions 7 to 10 use the same information.
A market researcher wishes to assess consumers’ preference among four different colours available on a name-brand dishwasher. In a sample of 198 recent sales, 61 were for stainless steel, 55 for white, 41 for black and 41 for cream.
To test the hypothesis that the preferences are in the ratio 6:5:4:3, a chi-square test is used. Write an expression for the P-value. Exactly one option must be correct)
a)
P-value = P(χ32 2.52) > 0.1
b)
P-value = P(χ32 8) < 0.05
c)
P-value = P(χ32 0.495) > 0.9
d)
P-value = P(χ1972 2.52) > .99

Choice (a) is correct!
Under this model, the Ei are respectively 66, 55, 44, 33 and τobs = 2.52. Since there are 4 categories, the appropriate chi-square is χ32 and the P-value is P(χ32 2.52) > 0.1.
Choice (b) is incorrect
Check your calculation of τobs.
Choice (c) is incorrect
Check your calculation of τobs.
Choice (d) is incorrect
You have used the wrong chi-square distribution.