Chapter 1: Hypothesis testing
PS
5 A manufacturer sells bags of 20 marbles in mixed colours. It claims that 30% of the marbles are red. Ginny thinks this is incorrect and tests the claim by opening a bag of marbles and counting how many are red. a Carry out a hypothesis test at the 10% level of significance given that Ginny finds three red marbles. Find the critical value for this test. b Suppose Ginny thinks the percentage should be lower than 30%. State the hypotheses. Will Ginny’s conclusion change? 6 It is claimed that the proportion of people worldwide with mixed-handedness (people who swap between hands to perform different tasks) is 1%. Amie thinks this proportion is incorrect. She interviews 600 people, and finds that 11 of them are mixed-handed. Test the claim at the 4% significance level.
E
PS
PL
EXPLORE 1.4 Exploring hypothesis testing
A hypothesis test involves a null hypothesis, an alternative hypothesis, a level of significance, calculation of the test statistic and a conclusion. This may appear to be a straightforward algorithm for reaching a decision about whether results are statistically significant; however, examples used in textbook explanations traditionally give the data alongside the hypotheses and a problem to be addressed. Discuss the following questions and explore whether the results of using this algorithm to answer hypothesis test questions are statistically meaningful. Were the data collected chosen at random? Why is this important?
Were the data collected before the test was set-up, or after the test was set-up? Why is this important?
SA M
• • •
Why is it important to decide upon the significance level before collecting the data?
Write an algorithm for a statistician to follow when setting up a hypothesis test.
1.3 Type I and Type II errors
We determine a population parameter from a complete set of data from that population, whereas a sample statistic is calculated from a subset of data from the complete population. When making inferences about a sample of data, such as conclusions from hypothesis testing, we use the sample statistic as if it is a known population parameter. The null and alternative hypotheses are set up with reference to a population parameter and conclusions made from calculations on a sample of observations. Consequently, there are two ways to reach an incorrect decision based on the results of calculating probabilities in hypothesis testing, a Type I error and a Type II error. The table shows how these errors arise. Accept H o
Reject H o
The null hypothesis, H o is true
Correct decision
Type I error
The null hypothesis, H o is false
Type II error
Correct decision
For example, in England and many other countries the law states that a person is innocent until ‘proven’ guilty. A Type I error happens when an innocent person is found guilty; a Type II error happens when a guilty person is found innocent. In both these situations, the evidence was either incorrect or not interpreted correctly.
Original material © Cambridge University Press 2017
17