If $P\left({\chi}_{5}^{2}>a\right)=0.1$,
find $a$.
Exactly one option must be correct)

*Choice (a) is correct!*

Use the chi-square tables with $\nu =5$,
and $p=.1$.

*Choice (b) is incorrect*

$P\left({\chi}_{5}^{2}>1.610\right)=0.9$.

*Choice (c) is incorrect*

You have used the
wrong value for $\nu $.

*Choice (d) is incorrect*

$P\left({\chi}_{5}^{2}>15.086\right)=0.01$.

Provide bounds for $p=P\left({\chi}_{25}^{2}>38.5\right)$.
Exactly one option must be correct)

*Choice (a) is incorrect*

Try again.

*Choice (b) is incorrect*

You have reversed the upper and lower limits. Obviously
$P\left({\chi}_{25}^{2}>38.5\right)$ 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, $P\left({\chi}_{25}^{2}>40.646\right)=0.025$ and
$P\left({\chi}_{25}^{2}>37.652\right)=0.05.$ Mark both 37.652 and
40.646 on the $x$-axis as well,
and it will be obvious that $P\left({\chi}_{25}^{2}>38.5\right)$
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, $P\left({\chi}_{25}^{2}>37.652\right)=0.05.$
Now mark 37.652 on your diagram, and it will be obvious that
$P\left({\chi}_{25}^{2}>38.5\right)$ 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 ${O}_{i}$: 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: Exactly one option must be correct)

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: Exactly one option must be correct)

*Choice (a) is incorrect*

The frequencies must satisfy ${\sum}_{i}{E}_{i}={\sum}_{i}{O}_{i}$.

*Choice (b) is correct!*

Under the model, the proportions for A, B and C are respectively
$\frac{1}{4},\phantom{\rule{1em}{0ex}}\frac{1}{2},\phantom{\rule{1em}{0ex}}\frac{1}{4}$.
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 ${O}_{i}$: 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 ${\tau}_{obs}={\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$, the observed value of the chi-square test statistic. Exactly one option must be correct)

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 ${\tau}_{obs}={\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$, the observed value of the chi-square test statistic. Exactly one option must be correct)

*Choice (a) is incorrect*

You have
found ${\sum}_{i}\frac{{O}_{i}^{2}}{{E}_{i}}$.

*Choice (b) is correct!*

${\tau}_{obs}=\frac{{\left(18-25\right)}^{2}}{25}+\frac{{\left(55-50\right)}^{2}}{50}+\frac{{\left(27-25\right)}^{2}}{25}=2.62$.

*Choice (c) is incorrect*

Check your calculations.

*Choice (d) is incorrect*

${\tau}_{obs}={\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$. You have
found instead, $\frac{{\sum}_{i}{\left({O}_{i}-{E}_{i}\right)}^{2}}{{\sum}_{i}{E}_{i}}$.

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, ${\chi}_{\nu}^{2}$ , for testing the goodness of fit of this model. Exactly one option must be correct)

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, ${\chi}_{\nu}^{2}$ , for testing the goodness of fit of this model. Exactly one option must be correct)

*Choice (a) is correct!*

There are three
categories, so $\nu =2$.

*Choice (b) is incorrect*

Try again. There
are three categories

*Choice (c) is incorrect*

$\nu =k-1$ 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 ${O}_{i}$: 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. Exactly one option must be correct)

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. Exactly one option must be correct)

*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
${\chi}_{2}^{2}$.

*Choice (d) is correct!*

Since there are 3 categories and since
${\tau}_{obs}=2.62$, the appropriate
chi-square distribution is ${\chi}_{2}^{2}$
and P-value = $P\left({\chi}_{2}^{2}\ge 2.62\right)>0.1$.
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 ${E}_{i}$ for the corresponding expected frequencies under the model, calculate ${\tau}_{obs}={\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$, the observed value of the chi-square test statistic. Exactly one option must be correct)

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 ${\tau}_{obs}={\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$, the observed value of the chi-square test statistic. Exactly one option must be correct)

*Choice (a) is correct!*

${\tau}_{obs}=\frac{{\left(61-49.5\right)}^{2}}{49.5}+\frac{{\left(55-49.5\right)}^{2}}{49.5}+\frac{{\left(41-49.5\right)}^{2}}{49.5}+\frac{{\left(41-49.5\right)}^{2}}{49.5}=6.2$.

*Choice (b) is incorrect*

Check your
calculations.

*Choice (c) is incorrect*

Check your calculations.

*Choice (d) is incorrect*

You have
found ${\sum}_{i}\frac{{O}_{i}^{2}}{{E}_{i}}$
instead of ${\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$

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 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)

*Choice (a) is incorrect*

You have used the wrong chi-square distribution.

*Choice (b) is correct!*

Under this
model, the ${E}_{i}$ are
all 49.5, and ${\tau}_{obs}=6.2$.
Since there are 4 categories, the appropriate chi-square is
${\chi}_{3}^{2}$, and P-value
= $P\left({\chi}_{3}^{2}\ge 6.2\right)>0.1$

*Choice (c) is incorrect*

Check your
calculation of ${\tau}_{obs}$.

*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 ${E}_{i}$ for the expected frequencies under a model which states that the preferences are in the ratio 6:5:4:3, calculate ${\tau}_{obs}={\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$, the observed value of the chi-square test statistic. Exactly one option must be correct)

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 ${E}_{i}$ for the expected frequencies under a model which states that the preferences are in the ratio 6:5:4:3, calculate ${\tau}_{obs}={\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$, the observed value of the chi-square test statistic. Exactly one option must be correct)

*Choice (a) is incorrect*

You have found ${\sum}_{i}\frac{{O}_{i}^{2}}{{E}_{i}}$
instead of ${\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$

*Choice (b) is incorrect*

The
expected frequencies under this model are 66, 55, 44, 33. Using these, recalculate
${\tau}_{obs}$.

*Choice (c) is incorrect*

${\tau}_{obs}={\sum}_{i}\frac{{\left({O}_{i}-{E}_{i}\right)}^{2}}{{E}_{i}}$. You have
found instead, $\frac{{\sum}_{i}{\left({O}_{i}-{E}_{i}\right)}^{2}}{{\sum}_{i}{E}_{i}}$.

*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
$\frac{6}{6+5+4+3}=\frac{1}{3}$
Multiplying by 198 gives 66. similarly for the other three colours.

Thus, ${\tau}_{obs}=\frac{{\left(61-66\right)}^{2}}{66}+\frac{{\left(55-55\right)}^{2}}{55}+\frac{{\left(41-44\right)}^{2}}{44}+\frac{{\left(41-33\right)}^{2}}{33}=2.52$.

Thus, ${\tau}_{obs}=\frac{{\left(61-66\right)}^{2}}{66}+\frac{{\left(55-55\right)}^{2}}{55}+\frac{{\left(41-44\right)}^{2}}{44}+\frac{{\left(41-33\right)}^{2}}{33}=2.52$.

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 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)

*Choice (a) is correct!*

Under this model,
the ${E}_{i}$ are respectively
66, 55, 44, 33 and ${\tau}_{obs}=2.52$.
Since there are 4 categories, the appropriate chi-square is
${\chi}_{3}^{2}$ and the P-value
is $P\left({\chi}_{3}^{2}\ge 2.52\right)>0.1$.

*Choice (b) is incorrect*

Check your
calculation of ${\tau}_{obs}$.

*Choice (c) is incorrect*

Check
your calculation of ${\tau}_{obs}$.

*Choice (d) is incorrect*

You have used the wrong chi-square distribution.