## MATH1015 Quizzes

Quiz 8: Binomial tests of significance
Question 1 Questions
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)
 a) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p>0.5$ b) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p<0.5$ c) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.5$ d) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p\ne 0.5$

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)
 a) ${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p>0.5$ b) ${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.5$ c) ${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p=0.5$ d) ${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p\ne 0.5$

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)
 a) Large values of $X$. b) Small values of $X$. c) Large and small values of $X$. d) There is insufficient information to determine this.

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)
 a) $P\left(X\le 4\right)=0.0095$ b) $P\left(X<4\right)=0.0017$ c) $P\left(X>4\right)=0.9905$ d) $P\left(X\ge 4\right)=0.9983$

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 $P\left(X\le 4\right)=0.0095$.
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) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=\frac{2}{15}$${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p\ne \frac{2}{15}$ b) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.07$${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p\ne 0.07$ c) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=\frac{2}{15}$${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p>\frac{2}{15}$ d) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.07$${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p>0.07$

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) $X\sim B\left(15,0.07\right)$ b) $X\sim B\left(15,0.93\right)$ c) $X\sim N\left(1.05,0.9765\right)$ d) $X\sim N\left(1.05,0.0043\right)$

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)
 a) The null hypothesis is true. b) The null hypothesis is false. c) The data are consistent with the null hypothesis. d) There is strong evidence against the null hypothesis.

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)
 a) The null hypothesis is true. b) The null hypothesis is false. c) There is no evidence against the null hypothesis. d) There is strong evidence against the null hypothesis.

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) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.012$${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.012$ b) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.012$${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p\ne 0.012$ c) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.02$${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p<0.02$ d) ${H}_{0}^{}:\phantom{\rule{1em}{0ex}}p=0.012$${H}_{1}^{}:\phantom{\rule{1em}{0ex}}p\ne 0.012$

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) The $p$-value is less than 0.05 and we have evidence to suggest that the new process lowers the failure rate. b) The $p$-value is less than 0.05 so the failure rate is lower than 0.02. c) The $p$-value is greater than 0.05 so there insufficient evidence to suggest that the new process lowers the failure rate. d) The $p$-value is greater than 0.05 so the failure rate is not than 0.02.

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.
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.