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University of Sydney
School of Mathematics
and Statistics
© Copyright 2004-2006

Quiz 8: Binomial tests of significance

Question

Question 1

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?

Not correct. Choice (a) is false.

Try again, this could only be an alternative hypothesis.

Not correct. Choice (b) is false.

Try again, this could only be an alternative hypothesis.

Your answer is correct.

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

Not correct. Choice (d) is false.

Try again, this could only be an alternative hypothesis.

Question 2

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?

Not correct. Choice (a) is false.

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

Not correct. Choice (b) is false.

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

Not correct. Choice (c) is false.

Try again. This is the null hypothesis.

Your answer is correct.

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

Question 3

Questions 3 and 4 use the following information.

Suppose we are conducting a test of significance where
$H0 : p = 0.7$
$H1 : p < 0.7$
$X$ is the number of successes and $X ~ B(12,0.7)$ when $H0$ is true.
Which observed values of $X$ argue against $H0$ in favour of $H1$ ?

Not correct. Choice (a) is false.

Try again. Large values would argue for $H1 : p > 0.7$ .

Your answer is correct.

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

Not correct. Choice (c) is false.

Try again. Large and small values would argue for $H1 : p ⁄= 0.7$

Not correct. Choice (d) is false.

Try again, we have sufficient information.

Question 4

Questions 3 and 4 use the following information.

Suppose we are conducting a test of significance where
$H0 : p = 0.7$
$H1 : p < 0.7$
$X$ is the number of successes and $X ~ B(12,0.7)$ when $H0$ is true.
Suppose the observed value of $X$ is 4. Which of the following is the $p$ -value?

Your answer is correct.

We are looking for small values of $X$ so p-value = $P (X ≤ 4).$
Using the tables for $X ~ B(12,0.7)$ we see $P(X ≤ 4) = 0.0095$ .

Not correct. Choice (b) is false.

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.

Not correct. Choice (c) is false.

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

Not correct. Choice (d) is false.

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

Question 5

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?

Not correct. Choice (a) is false.

Try again, $125$ is the observed probability of having the disorder.

Not correct. Choice (b) is false.

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

Not correct. Choice (c) is false.

Try again, $125$ is the observed probability of having the disorder.

Your answer 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 $H1 : p > 0.07$ .

Question 6

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 $H0$ is true what is the sampling distribution of $X$ ?

Your answer is correct.

If H₀ 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 $X ~ B(n, p) = B(15,0.07)$

Not correct. Choice (b) is false.

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

Not correct. Choice (c) is false.

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

Not correct. Choice (d) is false.

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

Question 7

On conducting a test of significance the $p$ -value is 0.1.
Which of the following statements is true?

Not correct. Choice (a) is false.

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

Not correct. Choice (b) is false.

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

Your answer 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.

Not correct. Choice (d) is false.

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.

Question 8

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?

Not correct. Choice (a) is false.

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

Not correct. Choice (b) is false.

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

Not correct. Choice (c) is false.

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.

Your answer 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.

Question 9

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.

Not correct. Choice (a) is false.

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

Not correct. Choice (b) is false.

Try again, 0.012 is the observed probability of faulty chip

Your answer is correct.

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

Not correct. Choice (d) is false.

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

Question 10

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?

Your answer is correct.

$H0 : p = 0.02$
$H1 : p < 0.02$
A suitable test statistic is $τ = X$ , the number of faulty chips and $X ~ B(1000,0.02)$ .
Small observed values of $X$ argue against $H0$ in favour of $H1$ .
The observed value is 12.
$X$ is approximately normal $Y ~ N(20,19.6)$ .
$12.5--20 P (X ≤ 12) ≈ P(Y < 12.5) = P(Z < √19.6 ) = P (Z < - 1.69) = 1- Φ(1.69) = 0.046 .$
The $p$ -value is less than 0.05. There is evidence to suggest that the failure rate has been lowered.

Not correct. Choice (b) is false.

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.

Not correct. Choice (c) is false.

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

Not correct. Choice (d) is false.

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.