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Quiz 12: Chi-square tests

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Question 1

If P(χ52 > a) = 0.1, find a.

a)
a = 9.236
   b)
a = 1.610
c)
a = 7.779
   d)
a = 15.086

 

Your answer is correct.
Use the chi-square tables with ν = 5, and p = .1.
Not correct. Choice (b) is false.
P(χ52 > 1.610) = 0.9.
Not correct. Choice (c) is false.
You have used the wrong value for ν.
Not correct. Choice (d) is false.
P(χ52 > 15.086) = 0.01.

Question 2

Provide bounds for p = P(χ252 > 38.5).

a)
0 ≤ p ≤ 0.01
   b)
0.05 ≤ p ≤ 0.025
c)
0.025 ≤ p ≤ 0.05
   d)
0.05 ≤ p ≤ 0.1

 

Not correct. Choice (a) is false.
Try again.
Not correct. Choice (b) is false.
You have reversed the upper and lower limits. Obviously P(χ252 > 38.5) must exceed 0.025.
Your answer 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).
Not correct. Choice (d) is false.
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.

Question 3

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:

a)
Ei: 10, 20, 10
   b)
Ei: 25, 50, 25
c)
Ei: 25, 25, 50
   d)
Ei: 50, 25, 25

 

Not correct. Choice (a) is false.
The frequencies must satisfy ∑ iEi = ∑ iOi.
Your answer is correct.
Under the model, the proportions for A, B and C are respectively 14, 12, 14. Multiplying by 100 gives the expected frequencies 25, 50, 25.
Not correct. Choice (c) is false.
These are not in the correct order.
Not correct. Choice (d) is false.
These are not in the correct order.

Question 4

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(O -E )2 --iEii-, the observed value of the chi-square test statistic.

a)
102.62
   b)
2.62
c)
0.262
   d)
0.78

 

Not correct. Choice (a) is false.
You have found ∑i O2i Ei .
Your answer is correct.
τobs = (18-25)2+ (55-50)2-+ (27-25)2-= 2.62 25 50 25 .
Not correct. Choice (c) is false.
Check your calculations.
Not correct. Choice (d) is false.
τ = ∑ (Oi--Ei)2 obs i Ei . You have found instead, ∑ --i(∑Oi-Ei)2 iEi .

Question 5

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, χ2ν , for testing the goodness of fit of this model.

a)
χ22
  b)
χ21
c)
χ23
  d)
χ299

 

Your answer is correct.
There are three categories, so ν = 2.
Not correct. Choice (b) is false.
Try again. There are three categories
Not correct. Choice (c) is false.
ν = k - 1 where k is the number of categories.
Not correct. Choice (d) is false.
There are 3 categories, not 100.

Question 6

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.

a)
P-value = P (χ2 ≥ 2.62) > 0.99. 99 The model is a good fit.
b)
P-value = P (χ2 ≥ 2.62) < 0.1. 1 The data provide evidence against the model.
c)
P-value = P (χ2 ≤ 2.62) 2 is very large. The data are consistent with the model.
d)
P-value = P (χ2≥ 2.62) > 0.1. 2 The data are consistent with the model.

 

Not correct. Choice (a) is false.
You have used the wrong chi-square distribution.
Not correct. Choice (b) is false.
You have used the wrong chi-square distribution.
Not correct. Choice (c) is false.
The P-value is not the lower tail of χ22.
Your answer is correct.
Since there are 3 categories and since τobs = 2.62, the appropriate chi-square distribution is χ22 and P-value = P (χ2 ≥ 2.62) > 0.1 2 . This means that the data are fairly typical of what is expected under the model.

Question 7

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(Oi-Ei)2 Ei, the observed value of the chi-square test statistic.

a)
τobs = 6.2
  b)
τobs = 10.3
c)
τobs = 206.7
  d)
τobs = 204.2

 

Your answer is correct.
2 2 2 2 τobs = (61-4499.5.5)-+ (55-4949..55)-+ (41-4949..55)-+ (41-449.9.55)-= 6.2 .
Not correct. Choice (b) is false.
Check your calculations.
Not correct. Choice (c) is false.
Check your caluclations.
Not correct. Choice (d) is false.
You have found 2 ∑i OEi i instead of ∑i (Oi-EEi)2 i

Question 8

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.

a)
P-value = 2 P (χ 2 ≥ 6.2) > 0.05
  b)
P-value = 2 P (χ3 ≥ 6.2) > 0.1
c)
P-value = 2 P (χ 3 ≥ 10.3) < 0.025.
  d)
P-value = P (χ2 ≥ 6.2) > 0.99 197

 

Not correct. Choice (a) is false.
You have used the wrong chi-square distribution.
Your answer 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 2 χ 3 , and P-value = 2 P (χ3 ≥ 6.2) > 0.1
Not correct. Choice (c) is false.
Check your caluclation of τobs.
Not correct. Choice (d) is false.
Try again. You have used the wrong chi-square distribution.

Question 9

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(Oi-EEi)2- i, the observed value of the chi-square test statistic.

a)
τobs = 200.52
  b)
τ = 8 obs
c)
τobs = 0.495
  d)
τobs = 2.52

 

Not correct. Choice (a) is false.
You have found ∑ O2i iEi instead of ∑ (Oi-Ei)2 i Ei
Not correct. Choice (b) is false.
The expected frequencies under this model are 66, 55, 44, 33. Using these, recalculate τobs.
Not correct. Choice (c) is false.
∑ (Oi-Ei)2- τobs = i Ei . You have found instead, ∑i(Oi-Ei)2 ∑iEi .
Your answer 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--- 1 6+5+4+3 = 3 Multiplying by 198 gives 66. similarly for the other three colours.
Thus, (61-66)2 (55-55)2 (41- 44)2 (41- 33)2 τobs =---66--+ ---55---+ ---44---+ ---33---= 2.52 .

Question 10

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.

a)
P-value = P (χ2 ≥ 2.52) > 0.1 3
  b)
P-value = P (χ2 ≥ 8) < 0.05 3
c)
P-value = P (χ2 ≥ 0.495) > 0.9 3
  d)
P-value = P (χ2 ≥ 2.52) > .99 197

 

Your answer is correct.
Under this model, the Ei are respectively 66, 55, 44, 33 and τ = 2.52 obs . Since there are 4 categories, the appropriate chi-square is χ2 3 and the P-value is P (χ2 ≥ 2.52) > 0.1 3 .
Not correct. Choice (b) is false.
Check your caluclation of τobs.
Not correct. Choice (c) is false.
Check your caluclation of τobs.
Not correct. Choice (d) is false.
You have used the wrong chi-square distribution.
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