Hypothesis testing is one of the most widely used procedures in statistics today, so we will spend more than one chapter discussing it. What is a hypothesis test?
The basic idea behind hypothesis testing is to assume that a population parameter equals some value (the null hypothesis), draw a sample and calculate a statistic, then use sampling distributions concepts to determine the likelihood of drawing such a sample, given the assumption of the parameter value.
Suppose we draw a random sample from the population and get a sample mean higher than the population mean. How far it is from the population mean? If \(\mu\)> is true, what is the probability that we would get this value? Is it so different from our hypothesized mean that our hypothesis cannot be true? We have to decide what “so different” means. Our sampling distribution tells us that the mean of all possible sample means should be the same as \(\mu\)>, so we can use those probability concepts to decide whether we think this sample statistic is extreme enough from the hypothesized mean to reject the null hypothesis.
The purpose of hypothesis statements is to draw a conclusion about population parameters, for which we do not have complete information.
Example10.1.1.
We want to say something about the average number of hours students at UCCS work per week. Let’s say that someone claims that this average is
If we survey \(2000\) students on campus, and in this sample the average number of hours worked per week was \(20\text{,}\) would you believe the claim that \(\mu=13.5\text{?}\)
If we survey \(10\) students on campus, and in this sample the average number of hours worked was \(14\text{,}\) would you believe the claim that \(13.5\text{?}\)
We would like to be able to use all the tools we’ve learned to confidently make statements about a population parameter based on a sample.
Definition10.1.2.
A hypothesis is an assumption about a population parameter.
The null hypothesis, denoted \(H_0\text{,}\) represents the status quo and involves stating the belief that the population parameter is \(=,\leq, \geq\) a specific value
The alternative hypothesis, denoted \(H_1\text{,}\) represents the opposite of \(H_0\text{,}\) and is believed to be true if the null hypothesis is found to be false.
Example10.1.3.
\(H_0\text{:}\) No more than \(30\%\) of the registered voters in Santa Clara County voted in the primary election.
\(H_1\text{:}\) More than \(30\%\) of the registered voters in Santa Clara County voted in the primary election.
Example10.1.4.
We want to test whether the mean number of hours UCCS students work per week is \(13.5\text{.}\) State the null and alternative hypotheses.
Answer.
\(H_0\text{:}\)\(\mu=13.5\) hours/week
\(H_1\text{:}\)\(\mu\neq 13.5\) hours/week
The purpose of hypothesis statements is to draw a conclusion about population parameters.
Definition10.1.5.
A one-tail hypothesis test has an alternative hypothesis stated as \(\lt \) or \(\gt\text{.}\)
A two-tail hypothesis test has an alternative hypothesis expressed as \(\neq\text{.}\)
Definition10.1.6.
A one-tail hypothesis test can be viewed as either a left-tail or right-tail test.
Figure10.1.7.Left-tail test powered by DesmosFigure10.1.8.Right-tail test powered by Desmos
Exercise10.1.9.
Look back at the problems mentioned below and decide if each is a one-tail or two-tail test.
Identify the null and alternative hypotheses for each scenario. Which involve a one-tail test and which involve a two-tail test?
(a)
In an effort to increase the number of people who file their returns electronically, suppose the Internal Revenue Service (IRS) launches a promotional campaign on the benefits of this filing method. As a follow-up to the campaign’s effectiveness, the IRS would like to test if the proportion of people who plan to be “e-filers” for the next tax season will exceed \(70\%\text{.}\)
Answer.
\(H_0\text{:}\)\(p\leq 0.7\)
\(H_1\text{:}\)\(p\gt 0.7\)
(one-tail/right-tail test)
(b)
To plan properly for equipment and services during the upcoming year, Comcast would like to test the hypothesis that the average number of televisions in the homes of its customers is equal to 2.9.
Answer.
\(H_0\text{:}\)\(\mu=2.9\)
\(H_1\text{:}\)\(\mu\neq 2.9\)
(two-tail test)
(c)
The federal government would like to determine the effectiveness of a recent tax-break program for first-time home buyers. Prior to the tax break, the average time period a house was on the market was 60 days. The government would like to test the claim that the current average time on the market is less than 60 days.
