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Quiz 12: Chi-square tests

Question

Question 1

P (χ_{5}^{2} > a) = 0.1

, find

a

Your answer is correct.

Use the chi-square tables with

ν = 5

, and

p = . 1

Not correct. Choice (b) is false.

P (χ_{5}^{2} > 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 (χ_{5}^{2} > 15.086) = 0.01

Question 2

Provide bounds for

p = P (χ_{25}^{2} > 38.5)

Not correct. Choice (a) is false.

Try again.

Not correct. Choice (b) is false.

You have reversed the upper and lower limits. Obviously

P (χ_{25}^{2} > 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 (χ_{25}^{2} > 40.646) = 0.025

and

P (χ_{25}^{2} > 37.652) = 0.05 .

Mark both 37.652 and 40.646 on the

x

-axis as well, and it will be obvious that

P (χ_{25}^{2} > 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 (χ_{25}^{2} > 37.652) = 0.05 .

Now mark 37.652 on your diagram, and it will be obvious that

P (χ_{25}^{2} > 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

O_{i}

: 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 (

E_{i}

) are:

Not correct. Choice (a) is false.

The frequencies must satisfy

\sum_{i} E_{i} = \sum_{i} O_{i}

Your answer is correct.

Under the model, the proportions for A, B and C are respectively

\frac{1}{4}, \frac{1}{2}, \frac{1}{4}

. 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

O_{i}

: 18, 55 and 27 respectively. A model states that A, B and C occur in the ratio 1:2:1.
Writing

E_{i}

for the corresponding expected frequencies under the model, calculate

τ_{o b s} = \sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

, the observed value of the chi-square test statistic.

Not correct. Choice (a) is false.

You have found

\sum_{i} \frac{O_{i}^{2}}{E_{i}}

Your answer is correct.

τ_{o b s} = \frac{{(18 - 25)}^{2}}{25} + \frac{{(55 - 50)}^{2}}{50} + \frac{{(27 - 25)}^{2}}{25} = 2.62

Not correct. Choice (c) is false.

Check your calculations.

Not correct. Choice (d) is false.

τ_{o b s} = \sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

. You have found instead,

\frac{\sum_{i} {(O_{i} - E_{i})}^{2}}{\sum_{i} E_{i}}

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

O_{i}

: 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,

χ_{ν}^{2}

, for testing the goodness of fit of this model.

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

O_{i}

: 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.

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

χ_{2}^{2}

Your answer is correct.

Since there are 3 categories and since

τ_{o b s} = 2.62

, the appropriate chi-square distribution is

χ_{2}^{2}

and P-value =

P (χ_{2}^{2} \geq 2.62) > 0.1

. 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

E_{i}

for the corresponding expected frequencies under the model, calculate

τ_{o b s} = \sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

, the observed value of the chi-square test statistic.

Your answer is correct.

τ_{o b s} = \frac{{(61 - 49.5)}^{2}}{49.5} + \frac{{(55 - 49.5)}^{2}}{49.5} + \frac{{(41 - 49.5)}^{2}}{49.5} + \frac{{(41 - 49.5)}^{2}}{49.5} = 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

\sum_{i} \frac{O_{i}^{2}}{E_{i}}

instead of

\sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

Question 8

Not correct. Choice (a) is false.

You have used the wrong chi-square distribution.

Your answer is correct.

Under this model, the

E_{i}

are all 49.5, and

τ_{o b s} = 6.2

. Since there are 4 categories, the appropriate chi-square is

χ_{3}^{2}

, and P-value =

P (χ_{3}^{2} \geq 6.2) > 0.1

Not correct. Choice (c) is false.

Check your caluclation of

τ_{o b s}

Not correct. Choice (d) is false.

Try again. You have used the wrong chi-square distribution.

Question 9

E_{i}

for the expected frequencies under a model which states that the preferences are in the ratio 6:5:4:3, calculate

τ_{o b s} = \sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

, the observed value of the chi-square test statistic.

Not correct. Choice (a) is false.

You have found

\sum_{i} \frac{O_{i}^{2}}{E_{i}}

instead of

\sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

Not correct. Choice (b) is false.

The expected frequencies under this model are 66, 55, 44, 33. Using these, recalculate

τ_{o b s}

Not correct. Choice (c) is false.

τ_{o b s} = \sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

. You have found instead,

\frac{\sum_{i} {(O_{i} - E_{i})}^{2}}{\sum_{i} E_{i}}

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

\frac{6}{6 + 5 + 4 + 3} = \frac{1}{3}

Multiplying by 198 gives 66. similarly for the other three colours.
Thus,

τ_{o b s} = \frac{{(61 - 66)}^{2}}{66} + \frac{{(55 - 55)}^{2}}{55} + \frac{{(41 - 44)}^{2}}{44} + \frac{{(41 - 33)}^{2}}{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.

Your answer is correct.

Under this model, the

E_{i}

are respectively 66, 55, 44, 33 and

τ_{o b s} = 2.52

. Since there are 4 categories, the appropriate chi-square is

χ_{3}^{2}

and the P-value is

P (χ_{3}^{2} \geq 2.52) > 0.1

Not correct. Choice (b) is false.

Check your caluclation of

τ_{o b s}

Not correct. Choice (c) is false.

Check your caluclation of

τ_{o b s}

Not correct. Choice (d) is false.

You have used the wrong chi-square distribution.

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Questions:

right first attempt
right
wrong

a)	$0 \leq p \leq 0.01$	b)	$0.05 \leq p \leq 0.025$
c)	$0.025 \leq p \leq 0.05$	d)	$0.05 \leq p \leq 0.1$

a)	$E_{i}$ : 10, 20, 10	b)	$E_{i}$ : 25, 50, 25
c)	$E_{i}$ : 25, 25, 50	d)	$E_{i}$ : 50, 25, 25

a)		P-value = $P (χ_{99}^{2} \geq 2.62) > 0.99 .$ The model is a good fit.
b)		P-value = $P (χ_{1}^{2} \geq 2.62) < 0.1 .$ The data provide evidence against the model.
c)		P-value = $P (χ_{2}^{2} \leq 2.62)$ is very large. The data are consistent with the model.
d)		P-value = $P (χ_{2}^{2} \geq 2.62) > 0.1 .$ The data are consistent with the model.

a)	$τ_{o b s} = 6.2$	b)	$τ_{o b s} = 10.3$
c)	$τ_{o b s} = 206.7$	d)	$τ_{o b s} = 204.2$

a)	P-value = $P (χ_{2}^{2} \geq 6.2) > 0.05$	b)	P-value = $P (χ_{3}^{2} \geq 6.2) > 0.1$
c)	P-value = $P (χ_{3}^{2} \geq 10.3) < 0.025 .$	d)	P-value = $P (χ_{197}^{2} \geq 6.2) > 0.99$

a)	$τ_{o b s} = 200.52$	b)	$τ_{o b s} = 8$
c)	$τ_{o b s} = 0.495$	d)	$τ_{o b s} = 2.52$

a)	P-value = $P (χ_{3}^{2} \geq 2.52) > 0.1$	b)	P-value = $P (χ_{3}^{2} \geq 8) < 0.05$
c)	P-value = $P (χ_{3}^{2} \geq 0.495) > 0.9$	d)	P-value = $P (χ_{197}^{2} \geq 2.52) > . 99$

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

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

a)	$χ_{2}^{2}$	b)	$χ_{1}^{2}$
c)	$χ_{3}^{2}$	d)	$χ_{99}^{2}$

The University of Sydney - School of Mathematics and Statistics

Quiz 12: Chi-square tests

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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