A coin is tossed 10 times. We wish to test the hypothesis that the coin is fair. Let
$p$
be the probability that the coin shows a head. Which of the following
represents the null hypothesis? Exactly one option must be correct)

*Choice (a) is incorrect*

Try again, this could only be an alternative hypothesis.

*Choice (b) is incorrect*

Try again, this could only be an alternative hypothesis.

*Choice (c) is correct!*

The probability that a fair coin shows a head is 0.5.

*Choice (d) is incorrect*

Try
again, this could only be an alternative hypothesis.

A coin is tossed 10 times. We wish to test the hypothesis that the coin is fair. Let
$p$
be the probability that the coin shows a head. Which of the following
represents the alternative hypothesis? Exactly one option must be correct)

*Choice (a) is incorrect*

Try again. This
would be the alternative hypothesis only if we wished to test that it is weighted in favour of heads.

*Choice (b) is incorrect*

Try again. This
would be the alternative hypothesis only if we wished to test that it is weighted in favour of tails.

*Choice (c) is incorrect*

Try again. This is
the null hypothesis.

*Choice (d) is correct!*

We need perform a two-sided test to test for fairness.

Questions 3 and 4 use the following information.

Suppose we are conducting a test of significance where

${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.7$

${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.7$

$X$ is the number of successes and $X\sim B\left(12,0.7\right)$ when ${H}_{0}^{}$ is true.

Which observed values of $X$ argue against ${H}_{0}^{}$ in favour of ${H}_{1}^{}$? Exactly one option must be correct)

Suppose we are conducting a test of significance where

${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.7$

${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.7$

$X$ is the number of successes and $X\sim B\left(12,0.7\right)$ when ${H}_{0}^{}$ is true.

Which observed values of $X$ argue against ${H}_{0}^{}$ in favour of ${H}_{1}^{}$? Exactly one option must be correct)

*Choice (a) is incorrect*

Try again. Large
values would argue for ${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p>0.7$.

*Choice (b) is correct!*

As the alternative hypothesis is suggesting a smaller number of successes, we are looking for small
values of $X$.

*Choice (c) is incorrect*

Try again. Large and small values would argue for
${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p\ne 0.7$

*Choice (d) is incorrect*

Try again, we have sufficient
information.

Questions 3 and 4 use the following information.

Suppose we are conducting a test of significance where

${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.7$

${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.7$

$X$ is the number of successes and $X\sim B\left(12,0.7\right)$ when ${H}_{0}^{}$ is true.

Suppose the observed value of $X$ is 4. Which of the following is the $p$-value? Exactly one option must be correct)

Suppose we are conducting a test of significance where

${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.7$

${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.7$

$X$ is the number of successes and $X\sim B\left(12,0.7\right)$ when ${H}_{0}^{}$ is true.

Suppose the observed value of $X$ is 4. Which of the following is the $p$-value? Exactly one option must be correct)

*Choice (a) is correct!*

We are looking for small values of $X$
so $p$-value
= $P\left(X\le 4\right)\phantom{\rule{0.3em}{0ex}}.$

Using the tables for $X\sim B\left(12,0.7\right)$ we see $P\left(X\le 4\right)=0.0095$.

Using the tables for $X\sim B\left(12,0.7\right)$ we see $P\left(X\le 4\right)=0.0095$.

*Choice (b) is incorrect*

Try again. The
$p$-value is the probability
of a result at least as extreme as the observed result, and always includes the observed result.

*Choice (c) is incorrect*

Try again, we are looking
for small values of $X$ so
we do not calculate $P\left(X>4\right)$.

*Choice (d) is incorrect*

Try again, we are looking
for small values of $X$ so
we do not calculate $P\left(X\ge 4\right)$.

Questions 5 and 6 use the following information.

A large health study has found that $7\%$ of the population suffers from a blood condition. A group of 15 people from an area near a mobile phone transmitter are tested for the condition and 2 people are found to have the condition. The locals believe that the transmitter increases the likelihood of having the condition. We wish to perform a test of significance on whether the mobile phone transmitter increases the incidence of the condition.

Let $p$ be the probability that an individual has the condition. Which of the following would be the appropriate null and alternative hypotheses? Exactly one option must be correct)

A large health study has found that $7\%$ of the population suffers from a blood condition. A group of 15 people from an area near a mobile phone transmitter are tested for the condition and 2 people are found to have the condition. The locals believe that the transmitter increases the likelihood of having the condition. We wish to perform a test of significance on whether the mobile phone transmitter increases the incidence of the condition.

Let $p$ be the probability that an individual has the condition. Which of the following would be the appropriate null and alternative hypotheses? Exactly one option must be correct)

*Choice (a) is incorrect*

Try
again, $\frac{2}{15}$
is the observed probability of having the disorder.

*Choice (b) is incorrect*

Try
again, this is a two sided test and we are testing for an increase in the incidence of the
disorder.

*Choice (c) is incorrect*

Try
again, $\frac{2}{15}$
is the observed probability of having the disorder.

*Choice (d) is correct!*

The large population
study tells us that $p=0.07$
and we are looking for an increase in incidence of the condition so
${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p>0.07$.

Questions 5 and 6 use the following information.

A large health study has found that $7\%$ of the population suffers from a blood condition. A group of 15 people from an area near a mobile phone transmitter are tested for the condition and 2 people are found to have the condition. The locals believe that the transmitter increases the likelihood of having the condition. We wish to perform a test of significance on whether the mobile phone transmitter increases the incidence of the condition.

Suppose $X$, the number of people with the condition, is an appropriate test statistic. When ${H}_{0}^{}$ is true what is the sampling distribution of $X$? Exactly one option must be correct)

A large health study has found that $7\%$ of the population suffers from a blood condition. A group of 15 people from an area near a mobile phone transmitter are tested for the condition and 2 people are found to have the condition. The locals believe that the transmitter increases the likelihood of having the condition. We wish to perform a test of significance on whether the mobile phone transmitter increases the incidence of the condition.

Suppose $X$, the number of people with the condition, is an appropriate test statistic. When ${H}_{0}^{}$ is true what is the sampling distribution of $X$? Exactly one option must be correct)

*Choice (a) is correct!*

If ${H}_{0}$ is true, the chance
of the condition is $p=0.07$,
and since $X$
counts the number with the condition in a sample of
$n=15$, the sampling
distribution of $X$
is $X\sim B\left(n,p\right)=B\left(15,0.07\right)$

*Choice (b) is incorrect*

Try
again, this is the sampling distribution for those who do not have the condition.

*Choice (c) is incorrect*

Try again, you cannot use a normal approximation to the binomial when
$n$ and
$p$ are this
small.

*Choice (d) is incorrect*

Try again, you cannot use a normal approximation to the binomial when
$n$ and
$p$ are
this small.

On conducting a test of significance the
$p$-value
is 0.1.

Which of the following statements is true? Exactly one option must be correct)

Which of the following statements is true? Exactly one option must be correct)

*Choice (a) is incorrect*

Try again, check the meaning of
$p$-value.

*Choice (b) is incorrect*

Try again, check the meaning of
$p$-value.

*Choice (c) is correct!*

There is a 10% chance of obtaining
the observed result or one more extreme if the null hypothesis is true. This suggests
the observed result is consistent with the null hypothesis.

*Choice (d) is incorrect*

Try again. A 10% chance of obtaining the observed result
or one more extreme if the null hypothesis is true, is not strong evidence against the
null hypothesis.

On conducting a test of significance the
$p$-value
is calculated and found to be less than 0.01.

Which of the following statements is true? Exactly one option must be correct)

Which of the following statements is true? Exactly one option must be correct)

*Choice (a) is incorrect*

Try again, check the meaning of
$p$-value.

*Choice (b) is incorrect*

Try again, check the meaning of
$p$-value.

*Choice (c) is incorrect*

Try again, a less than 1% chance of
obtaining the observed result or one more extreme if the null hypothesis is true is
evidence against the null hypothesis.

*Choice (d) is correct!*

There is a less than 1% chance of obtaining the observed result or one
more extreme if the null hypothesis is true. This is regarded as strong evidence
against the null hypothesis.

Questions 9 and 10 use the same information.

A computer chip manufacturer has a failure rate of 2% using the current technology. A new process has been developed to lower the failure rate. One thousand chips have been produced with the new process and there were 12 faulty chips. We wish to test whether the new process is better.

Let $p$ be the probability that a chip is faulty. Which of the following are the appropriate null and alternative hypotheses. Exactly one option must be correct)

A computer chip manufacturer has a failure rate of 2% using the current technology. A new process has been developed to lower the failure rate. One thousand chips have been produced with the new process and there were 12 faulty chips. We wish to test whether the new process is better.

Let $p$ be the probability that a chip is faulty. Which of the following are the appropriate null and alternative hypotheses. Exactly one option must be correct)

*Choice (a) is incorrect*

Try again, 0.012 is the observed probability of faulty chip.

*Choice (b) is incorrect*

Try again, 0.012 is the observed probability of faulty chip

*Choice (c) is correct!*

The
new process has been developed to lower the failure rate so a one-sided test can be
conducted.

*Choice (d) is incorrect*

Try
again, the new process has been developed to lower the failure rate so a one-sided
test should be conducted.

Questions 9 and 10 use the same information.

A computer chip manufacturer has a failure rate of 2% using the current technology. A new process has been developed to lower the failure rate. One thousand chips have been produced with the new process and there were 12 faulty chips. We wish to test whether the new process is better.

Using the null and alternative hypotheses from Question 9 and after conducting a test of significance which of the following statements is correct? Exactly one option must be correct)

A computer chip manufacturer has a failure rate of 2% using the current technology. A new process has been developed to lower the failure rate. One thousand chips have been produced with the new process and there were 12 faulty chips. We wish to test whether the new process is better.

Using the null and alternative hypotheses from Question 9 and after conducting a test of significance which of the following statements is correct? Exactly one option must be correct)

*Choice (a) is correct!*

${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.02$

${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.02$

A suitable test statistic is $\tau =X$, the number of faulty chips and $X\sim B\left(1000,0.02\right)$.

Small observed values of $X$ argue against ${H}_{0}^{}$ in favour of ${H}_{1}^{}$.

The observed value is 12.

$X$ is approximately normal $Y\sim N\left(20,19.6\right)$.

$P\left(X\le 12\right)\approx P\left(Y<12.5\right)=P\left(Z<\frac{12.5-20}{\sqrt{19.6}}\right)=P\left(Z<-1.69\right)=1-\Phi \left(1.69\right)=0.046\phantom{\rule{0.3em}{0ex}}.$

The $p$-value is less than 0.05. There is evidence to suggest that the failure rate has been lowered.

${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.02$

A suitable test statistic is $\tau =X$, the number of faulty chips and $X\sim B\left(1000,0.02\right)$.

Small observed values of $X$ argue against ${H}_{0}^{}$ in favour of ${H}_{1}^{}$.

The observed value is 12.

$X$ is approximately normal $Y\sim N\left(20,19.6\right)$.

$P\left(X\le 12\right)\approx P\left(Y<12.5\right)=P\left(Z<\frac{12.5-20}{\sqrt{19.6}}\right)=P\left(Z<-1.69\right)=1-\Phi \left(1.69\right)=0.046\phantom{\rule{0.3em}{0ex}}.$

The $p$-value is less than 0.05. There is evidence to suggest that the failure rate has been lowered.

*Choice (b) is incorrect*

Try again, a test of
significance can only tell us whether or not there is evidence that the
failure rate is lower than 0.02, not whether the claim is true or false.

*Choice (c) is incorrect*

Try again, the
$p$-value is less than
0.05

*Choice (d) is incorrect*

Try again, a test of
significance can only tell us whether or not there is evidence that the failure rate is
lower than 0.02, not whether the claim is true or false.