Topic 7.4: Setting Up a Test for a Population Mean
Null and Alternative Hypotheses, Procedure, and Conditions
Video Follow-Along Worksheet
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Learning Objectives:
State the null hypothesis for a significance test about a population mean.
State the alternative hypothesis that matches the question of interest.
Distinguish between one-sided and two-sided alternatives for tests about μ.
Identify a one-sample t-test as the correct procedure for testing a single population mean.
Check the randomization, 10%, and Nearly Normal conditions for a significance test for a population mean.
Essential Knowledge:
A significance test evaluates a claim about a population parameter and asks whether the observed evidence would be unusual if the null hypothesis were true.
The null hypothesis is written with equality using the population parameter μ and a hypothesized value.
The alternative hypothesis uses a strict inequality: less than, greater than, or not equal to.
The form of the alternative hypothesis is determined by the research question and should be chosen before data are collected.
Hypotheses should refer to the population mean, not a sample statistic such as x̄.
For a one-sample test about μ, check for a random sample or randomized experiment, verify the 10% condition when sampling without replacement, and use either a large sample or sample data with no strong skewness or outliers.
A procedure used to test a claim about a population parameter by asking whether the sample evidence is likely to occur by chance alone.
Null Hypothesis
The statement of equality that represents no difference, no change, or the claimed value for the population mean.
Alternative Hypothesis
The claim the data are intended to support; it uses a strict inequality.
Null Value
The hypothesized value that appears in both the null and alternative hypotheses.
One-Sided Alternative
An alternative hypothesis of the form μ < value or μ > value.
Two-Sided Alternative
An alternative hypothesis of the form μ ≠ value.
One-Sample t-Test
The inference procedure used to test a claim about one population mean when the population standard deviation is unknown.
Conditions for Inference
The assumptions that must be checked before carrying out the one-sample t-test.
Video 1: Stating the Null and Alternative Hypotheses
[0:00 – 7:07]
1.[0:15]
In this video, we learn how to state the hypothesis and the hypothesis in a significance test for a population mean.
2.[0:58]
A significance test is used to test a claim about the value of a population and decide whether the evidence is likely or unlikely to happen by alone.
3.[1:18]
In Unit 7, quantitative data are summarized using the sample , and Topics 7.4 and 7.5 focus on significance tests for a population mean.
Context — Got Hops?[1:54 – 2:31]
An online article claims that the average vertical jump for teens is 15 inches. A random sample of 20 students at one large high school had a mean jump of 15.8 inches with a standard deviation of 2.33 inches. The question is whether the average jump for students at this school differs from 15 inches.
4.[2:31]
The null hypothesis is often a claim of no or no .
5.[2:31]
For the vertical jump example, the null hypothesis is H0: μ = inches.
6.[3:20]
The alternative hypothesis is the claim we hope to support with evidence from the .
7.[3:20]
Because the question asks whether the mean jump differs from 15 inches, the alternative is Ha: μ 15.
8.[3:53]
In this example, μ is the mean vertical jump for all students at this high .
Hypothesis Forms
H0: μ = μ0
Ha: μ < μ0 or μ > μ0 or μ ≠ μ0
9.[4:12]
The null hypothesis is a statement of , while the alternative always contains a strict .
10.[4:44]
If the alternative uses < or >, it is . If it uses ≠, it is .
11.[5:01]
The choice of alternative should be stated before , and hypotheses should never use statistics like .
Practice — I Can Go for Miles and Miles[5:22 – 5:58]
A tire manufacturer tests whether Tread40 tires last more than 40,000 miles on average. A random sample of 35 tires had a mean mileage of 42,348 miles with a standard deviation of 2,140 miles.
12.[5:58]
For the tire example, the null hypothesis is H0: μ = miles.
13.[5:58]
Because the engineer wants to know if the tires last more than 40,000 miles, the alternative is μ > .
14.[6:40]
In context, μ is the mean mileage for all tires.
Video 2: Identifying the Procedure and Checking Conditions
[0:00 – 5:20]
15.[0:17]
In this video, we learn how to identify the correct significance test for a population mean and how to verify the .
Return to Got Hops?[0:34 – 1:25]
The same vertical jump example is used again. The hypotheses are H0: μ = 15 inches and Ha: μ ≠ 15 inches, where μ is the mean vertical jump for all students at the high school.
16.[1:25]
When the goal is to test a claim about a population mean, the correct procedure is a one-sample for a population mean.
Conditions Checklist for a One-Sample t-Test:[2:11]
Data come from a random sample or a randomized experiment.
If sampling without replacement, the sample size is less than 10% of the population.
If n < 30, show that the sample data have no strong skewness or outliers; if n ≥ 30, the large-sample condition is enough.
17.[2:11]
To check independence, confirm that the data came from a random sample or a randomized .
18.[2:11]
When sampling without replacement, the sample size should be less than of the population size.
19.[2:11]
The shape condition is satisfied if the sample size is at least , or if smaller samples show no strong or .
20.[3:08]
In the vertical jump example, the 20 students were randomly , 20 is less than of all students at a large high school, and the graph showed no strong skewness or outliers.
21.[3:08]
Because n is less than 30, you should actually make a of the sample data before saying the conditions are met.
Counterexample — Long Live the Tablet![4:10 – 5:20]
CB Tablets claims its tablets have an average battery life of 14 hours. A consumer group wonders whether the average battery life is shorter than 14 hours. In the sample of 10 tablets, there was no indication the tablets were randomly selected, and a dotplot showed strong skewness with a potential outlier.
22.[4:10]
In the tablet example, the conditions were because there was no evidence of a sample and the dotplot showed strong skewness with a potential .
Key Takeaways
[Summary]
Write hypotheses using the population mean μ, not the sample mean x̄.
The null hypothesis uses equality, and the alternative uses a strict inequality chosen to match the research question.
Use a one-sample t-test to test a claim about one population mean.
Before testing, verify randomization, the 10% condition, and an appropriate shape condition in context.
Reflection Questions
[AI-Graded]
R1.Using the Got Hops and Tread40 tire examples, explain how to state the null and alternative hypotheses for a test about a population mean. Include why the null uses equality, why the alternative uses an inequality, how the question determines whether the test is one-sided or two-sided, and what μ means in each context.
R2.Explain how to identify the correct procedure and check the conditions for a significance test for a population mean. Use the vertical jump example and the tablet example to show the difference between a situation where the conditions are met and one where they are not.
Exit Ticket
A tire manufacturer wants to test whether Tread40 tires last more than 40,000 miles on average. A random sample of 35 tires had a sample mean of 42,348 miles and a sample standard deviation of 2,140 miles.
(a) Define the parameter and state the null and alternative hypotheses in symbols and words.
(b) Identify the correct significance test procedure and explain why the conditions are reasonable.
(c) Explain why this is a one-sided test and why x̄ should not appear in the hypotheses.
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