1. Hypotheses tests for means
Chapter 23 – Part B
Hypotheses tests for means
Objective: To test claims about inferences for means, under specific conditions

2. The hypotheses for proportions are similar to those for proportions.
In fact, they are the same. We use µ instead of p.
H0: µ = 3.2

3. Reasoning of Hypothesis testing
There are four basic parts to a hypothesis test:
Hypotheses
Model
Mechanics
Conclusion
Let’s look at these parts in detail…

4. Reasoning of Hypothesis testing (cont.)
1. Hypotheses
The null hypothesis: To perform a hypothesis test, we must first translate our question of interest into a statement about model parameters.
In general, we have
H0: parameter = hypothesized value.
The alternative hypothesis: The alternative hypothesis, HA, contains the values of the parameter we consider plausible when we reject the null.

5. Reasoning of Hypothesis testing (cont.)
Model
To plan a statistical hypothesis test, specify the model you will use to test the null hypothesis and the parameter of interest.
All models require assumptions, so state the assumptions and check any corresponding conditions.
Your conditions should conclude with a statement such as:
Because the conditions are satisfied, I can model the sampling distribution of the proportion with a Normal model.
Watch out, though. It might be the case that your model step ends with “Because the conditions are not satisfied, I can’t proceed with the test.” If that’s the case, stop and reconsider (proceed with caution)

6. Reasoning of Hypothesis testing (cont.)
Model (cont.)
Don’t forget to name your test!
The test about means is called a one-sample t-test.

7. Reasoning of Hypothesis testing (cont.)
Model (cont.)
One-sample t-test for the mean
The conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval.
We test the hypothesis H0:  = 0 using the statistic
𝒕 𝒏−𝟏 = 𝒙 − 𝝁 𝟎 𝑺𝑬( 𝒙 )
The standard error of the sample mean is 𝑺𝑬 𝒙 = 𝒔 𝒏
When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t model with n – 1 df. We use that model to obtain a P- value.

8. Reasoning of Hypothesis testing (cont.)
Model (cont.)
Finding the P-Value
Either use the table provided, or you may use your calculator:
normalcdf( is used for z-scores (if you know 𝜎)
tcdf( is used for critical t-values (when you use s to estimate 𝜎)
2nd  Distribution
tcdf(lower bound, upper bound, degrees of freedom)

9. Reasoning of Hypothesis testing (cont.)
Mechanics
Under “mechanics” we place the actual calculation of our test statistic from the data.
Different tests will have different formulas and different test statistics.
Usually, the mechanics are handled by a statistics program or calculator, but it’s good to know the formulas.

10. Reasoning of Hypothesis testing (cont.)
Mechanics (continued)
The ultimate goal of the calculation is to obtain a P-value.
The P-value is the probability that the observed statistic value (or an even more extreme value) could occur if the null model were correct.
If the P-value is small enough, we’ll reject the null hypothesis.
Note: The P-value is a conditional probability—it’s the probability that the observed results could have happened if the null hypothesis is true.

11. Reasoning of Hypothesis testing (cont.)
Conclusion
The conclusion in a hypothesis test is always a statement about the null hypothesis.
The conclusion must state either that we reject or that we fail to reject the null hypothesis.
And, as always, the conclusion should be stated in context.

12. Steps for Hypothesis testing for one-sample t-test for means
Check Conditions and show that you have checked these!
Random Sample: Can we assume this?
10% Condition: Do you believe that your sample size is less than 10% of the population size?
Nearly Normal:
If you have raw data, graph a histogram to check to see if it is approximately symmetric and sketch the histogram on your paper.
If you do not have raw data, check to see if the problem states that the distribution is approximately Normal.

13. Steps for Hypothesis testing for one-sample t-test for means (cont.)
State the test you are about to conduct
Ex) One-Sample t-Test for Means
Set up your hypotheses
H0:
HA:
Calculate your test statistic
𝒕 𝒏−𝟏 = 𝒙 − 𝝁 𝟎 𝒔 𝒏
Draw a picture of your desired area under the t-model, and calculate your P-value.

14. Steps for Hypothesis testing for one-sample t-test for means (cont.)
Make your conclusion.
When your P-value is small enough (or below α, if given), reject the null hypothesis.
When your P-value is not small enough, fail to reject the null hypothesis.

15. Calculator tips
Given a set of data:
Enter data into L1
Set up STATPLOT to create a histogram to check the nearly Normal condition
STAT  TESTS  2:T-Test
Choose Stored Data, then specify your data list (usually L1)
Enter the mean of the null model and indicate where the data are (>, <, or ≠)
Given sample mean and standard deviation:
Choose Stats  enter
Specify the hypothesized mean and sample statistics
Specify the tail (>, <, or ≠)
Calculate

16. Example 1
A company has set a goal of developing a battery that lasts over 5 hours (300 minutes) in continuous use. A first test of 12 of these batteries measured the following lifespans (in minutes): 321, 295, 332, 351, 281, 336, 311, 253, 270, 326, 311, and 288. Is there evidence that the company has met its goal?

17. Example 1 (continued)
Find a 90% confidence interval for the mean lifespan of this type of battery.

18. Example 2 (Partners)
Cola makers test new recipes for loss of sweetness during storage. Trained tasters rate the sweetness before and after storage. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by 10 tasters for one new cola recipe: Are these data good evidence that the cola lost sweetness?

19. Day 2

20. Example 3
Psychology experiments sometimes involve testing the ability of rats to navigate mazes. The mazes are classified according to difficulty, as measured by the mean length of time it takes rats to find the food at the end. One researcher needs a maze that will take the rats an average of about one minutes to solve. He tests one maze on several rats, collecting the data shown. Test the hypothesis that the mean completion time for this maze is 60 seconds at an alpha level of What is your conclusion?
38.4
57.6
46.2
55.5
62.5
49.5
38.0
40.9
62.8
44.3
33.9
93.8
50.4
47.9
35.0
69.2
52.8
60.1
56.3
55.1

21. Confidence intervals & Hypothesis tests
Confidence intervals and hypothesis tests are built from the same calculations.
They have the same assumptions and conditions.
You can approximate a hypothesis test by examining a confidence interval.
Just ask whether the null hypothesis value is consistent with a confidence interval for the parameter at the corresponding confidence level.

22. Confidence intervals & Hypothesis tests (cont.)
Because confidence intervals are two-sided, they correspond to two-sided tests.
In general, a confidence interval with a confidence level of C% corresponds to a two-sided hypothesis test with an -level of
100 – C%.
The relationship between confidence intervals and one-sided hypothesis tests is a little more complicated.
A confidence interval with a confidence level of C% corresponds to a one-sided hypothesis test with an -level of ½(100 – C)%.

23. When we perform a hypothesis test, we can make mistakes in two ways:
Making Errors
Here’s some shocking news for you: nobody’s perfect. Even with lots of evidence we can still make the wrong decision.
When we perform a hypothesis test, we can make mistakes in two ways:
The null hypothesis is true, but we mistakenly reject it.
(Type I error)
II The null hypothesis is false, but we fail to reject it.
(Type II error)

24. Here’s an illustration of the four situations in a hypothesis test:
Making Errors (cont.)
Which type of error is more serious depends on the situation at hand. In other words, the importance of the error is context dependent.
Here’s an illustration of the four situations in a hypothesis test:

25. What type of error was made?
Making Errors (cont.)
What type of error was made?
How about OJ Simpson?

26. How often will a Type I error occur?
Making Errors (cont.)
How often will a Type I error occur?
A Type I error is rejecting a true null hypothesis. To reject the null hypothesis, the P-value must fall below . Therefore, when the null is true, that happens exactly with a probability of . Thus, the probability of a Type I error is our  level.
When H0 is false and we reject it, we have done the right thing.
A test’s ability to detect a false null hypothesis is called the power of the test.

27. We assign the letter  to the probability of this mistake.
Making Errors (cont.)
When H0 is false and we fail to reject it, we have made a Type II error.
We assign the letter  to the probability of this mistake.
It’s harder to assess the value of  because we don’t know what the value of the parameter really is.
When the null hypothesis is true, it specifies a single parameter value, H0: parameter = hypothesized value.
When the null hypothesis is false, we do not have a specific parameter; we have many possible values.
There is no single value for  --we can think of a whole collection of ’s, one for each incorrect parameter value.

28. Ask “How big a difference would matter?”
Making Errors (cont.)
One way to focus our attention on a particular  is to think about the effect size.
Ask “How big a difference would matter?”
We could reduce  for all alternative parameter values by increasing .
This would reduce  but increase the chance of a Type I error.
This tension between Type I and Type II errors is inevitable.
The only way to reduce both types of errors is to collect more data. Otherwise, we just wind up trading off one kind of error against the other.

29. When we calculate power, we imagine that the null hypothesis is false.
Power of the test
The power of a test is the probability that it correctly rejects a false null hypothesis.
The power of a test is 1 –  ; because  is the probability that a test fails to reject a false null hypothesis and power is the probability that it does reject.
Whenever a study fails to reject its null hypothesis, the test’s power comes into question.
When we calculate power, we imagine that the null hypothesis is false.

30. Power of the test (cont.)
The value of the power depends on how far the truth lies from the null hypothesis value.
The distance between the null hypothesis value, 0 , and the truth,  , is called the effect size.
Power depends directly on effect size. It is easier to see larger effects, so the farther  is from 0, the greater the power.

31. Assignments
Day 1: pp # 1 - 5
pp # 23, 24
Day 2: pp # 1cd, 2cd, 29, 30, 33, 35
Day 3: pp # 22, 25 – 28, 34