Reading sections are from the Introductory Statistics Textbook
Imagine you are in a courtroom as a member of the jury. The defendant comes in. As a member of the jury,
innocentinnocent verdict to guilty
guiltyinnocent assumption, so it remains.This is actually a good picture of what we are doing with hypothesis testing. We have some pre-detemined value which we call the null value.
We then gather data and use statistics to determine if the null value fits with the data or not.
Our default assumption in statistics is that the true value (the population mean for quantitative data) has not changed. It is the same as the null value. This is known as a Null Hypothesis.
\[H_0: \mu = \mu_0\text{ for Quantitative variables} \qquad\qquad H_0: p = p_0\text{ for Categorical variables}\]For problems within this course, the null value will often be given to you. In the real world, this may take some research. It may be published in a research paper, it may be a calculation from last year, or it may be something you need to find on your own. For this class, we’ll make it easy for you and give it to you.
Consider this problem we saw back in Lesson 18 (I’ll adapt the question to fit what we are looking for in hypothesis testing):
American mental abilities are often measured by an IQ test. The IQ distribution is normal with a mean of 100 and a population standard deviation of 15. A random sample of 40 Snow College students is taken and they have an average IQ of 106.3. Can we say with a 5% level of significance that the true average IQ of Snow College students is higher than the nation?
(For now, don’t worry about the 5% level of significance. We’ll address that in the next section).
In this case, we have a sample that represents all Snow College students (\(\bar{x} = 106.3\)). We don’t know what the population mean is (\(\mu\)). We can find a Confidence Interval to find this. But the question is how this population mean compares to the mean for the entire United States. Since we want to compare Snow College IQ scores to the national scores, the national average is our null value.
\[\mu_0 = 100\]So, our Null Hypothesis, or our default assumption, is that the average IQ of Snow College students is the same as for the nation. Specifically, we assume that the average IQ of Snow College students is 100.
\[H_0: \mu = 100\]What if the null hypothesis isn’t true? We need to establish a second hypothesis, known as an Alternate Hypothesis, which will be our assumption if the null hypothesis fails. The alternate hypothesis is based on what we want to show in our question. There are 3 different possible alternate hypothesis:
In the IQ question, we want to see whether the IQ of Snow College students is “higher than” the national average. So, we take the first option for our alternate hypothesis.
\[H_0: \mu = 100 \qquad\qquad H_A: \mu > 100\]The rest of the lesson will look at how to determine which hypothesis we accept.
For each of the following questions, establish both the Null Hypothesis and the Alternate Hypothesis
1. A coffee shop claims that the average wait time for a customer to receive their order is no more than 4 minutes. A customer advocacy group believes the wait time is longer. A random sample of 35 customers had an average wait time of 4.6 minutes with a standard deviation of 1.2 minutes. At the 5% significance level, test the claim that the average wait time is greater than 4 minutes.
Our study is comparing the average wait time to 4 minutes. This is our null value.
$$\mu_0 = 4$$The default assumption is that the true mean is 4 minutes. We are testing to see if the wait time at the coffee shop is longer than 4 minutes. So, our hypotheses are:
$$H_0: \mu = 4 \qquad \qquad H_A: \mu > 4$$2. A smartphone manufacturer advertises that their new model has an average battery life of 20 hours. A tech reviewer suspects the battery life is less than advertised. A sample of 55 phones showed an average battery life of 18.7 hours with a standard deviation of 2.5 hours. Conduct a hypothesis test at the 1% significance level.
The advertisement says the average is 20 hours, and we are comparing our phone batteries to this number, so this is our null value.
$$\mu_0 = 20$$The default assumption is that this is correct. The test is to see if the time is less than advertised. So, our hypotheses are:
$$H_0: \mu = 20 \qquad\qquad H_A: \mu < 20$$3. An educator wants to test whether a new teaching method affects student performance. Historically, the average score on a standardized exam is 75. The educator believes the new method may lead to a different average score. A class of 40 students taught with the new method had an average score of 78.2 with a standard deviation of 6.3. Test the hypothesis at the 5% significance level.
The standardized exam shows an average score of 75 using the previous teaching method. This is what we are comparing the new teaching method to, so we use 75 as our null value.
$$\mu_0 = 75$$We want to see if the new teaching method leads to different average scores. In this case, we don't care if the score is greater or smaller - just different. So, our hypotheses are:
$$H_0: \mu = 75 \qquad\qquad H_A: \mu \ne 75$$