Section 10.5: Putting It Together: Which Method Do I Use?
Objectives
By the end of this lesson, you will be able to...
- determine the appropriate hypothesis test to perform
Hypothesis Test Summary
So far this semester, we've learned three different hypothesis tests, based on the parameter of interest, and what information is given. Those three are:
Tests Regarding the Population Proportion
In order to perform a hypothesis test regarding the population proportion, all of the following must be true:
- the sample is a random sample, and
- the sample is less than 5% of the population (n≤0.05N), and
- np(1-p)≥10
The sample statistic for this test is:
Tests Regarding the Population Mean
In order to perform a hypothesis test regarding the population mean, the sample must be a random sample, and one of the following must be true:
- the sample was randomly selected, and
- the population from which the sample is drawn is normally distributed or the sample size is reasonably large (n≥30)
| Two-Tailed H0: μ = μ0 H1: μ ≠ μ0 |
Left-Tailed H0: μ = μ0 H1: μ < μ0 |
Right-Tailed H0: μ = μ0 H1: μ > μ0 |
The sample statistic for this test is:
with n-1 degrees of freedom.
Tests Regarding the Population Standard Deviation
In order to perform a hypothesis test regarding the population standard deviation, the sample must come from a normally distributed population. In this case, the sample statistic is:
Choosing the Appropriate Hypothesis Test
So now the big question is which test to apply. Here's a flowchart to represent the process:
Unfortunately, like most problems where you need to choose which technique to apply, there's no handy blueprint that you can always follow. The key, then, is to try as many examples as possible (like those on the next page) and doing all the assigned homework as soon as possible.
For each example, state null and alternative hypotheses and choose the appropriate test statistic.
Example 1
According to the US Census Bureau, approximately 42% of Americans ages 18-24 voted in the 2004 presidential election. We wonder if the percentage of ECC students is different from this. If we collect a simple random sample of 200 ECC students and find that 103 of them voted in the 2008 presidential election, is there enough evidence at the 10% level of significance to support our claim?
Even though the word percentage is used here, this is a hypothesis test regarding the population proportion.
H0: p = 0.42
H1: p ≠ 0.42
The test statistic is:
Example 2
According to the ECC website, the average age of ECC students is 28.2 years. Suppose we collect data from 30 Mth120 online students and find an average age of 20.3 years, with a standard deviation of 2.3 years.
Is there enough evidence at the 5% level of significance to support the claim that the average age of Mth120 online students is less than 28.2 years?
Since we're looking at average age, the parameter under consideration is the population mean. We have to be careful about the information we're given, though. The standard deviation shown is from the sample, not the population, so we'll have to use a t statistic, not a z.
H0: μ = 28.2 yrs
H1: μ < 28.2 yrs
And the test statistic would be: 
Example 3
We know from previous examples that the standard deviation of IQs is normally distributed with a standard deviation of 15. Suppose we wonder if the IQs of ECC students have more variation. To answer this question, we collect the IQs from a random sample of ECC students and find a standard deviation of 16.2.
Based on this information, is there enough evidence at the 5% level of significance to say that the IQs of ECC students have more variation than the general population?
The problem is talking about variation, which always implies variance or standard deviation. Also, we're given the standard deviation of the sample, which implies that this is a hypothesis test concerning the population standard deviation.
H0: σ = 15
H1: σ > 15
The test statistic is:
