1. Lecture #23 Tuesday, November 8, 2016 Textbook: 13.1 through 13.4
Statistics 200
Lecture #23 Tuesday, November 8, 2016
Textbook: through 13.4
Objectives:
• Formulate null and alternative hypotheses involving population means
• Calculate T-statistics (for means) instead of Z-statistics (for proportions); determine correct degrees of freedom.
• Determine correct test procedure from context of problem.
• Identify potential errors that can occur in hypothesis testing; distinguish between type I and type II errors.

2. Statistical Hypotheses
Null Hypothesis, H0:
Nothing happening
No change / difference
Alternative Hypothesis, Ha:
Something is happening
There is a change / difference

3. Five steps for a statistical hypothesis test
H0 and Ha
Z = ___ or T= ___
p = ____
Reject H0 or
fail to reject H0
We conclude that…
Formulate null and alternative hypotheses.
(Verify necessary data conditions and) summarize data into a test statistic.
Find a p-value.
Make a decision.
Report the conclusion in context.

4. General test statistic formula:
An example from Lecture 18: A test for one proportion
A sample of 300 drivers from the 16–24 age group found 105 who say that they have driven while drowsy in the last year. Have more than 30% of this age group driven while drowsy in the past year?
Check: Both n*p-hat and n*(1–p-hat) are 10.

5. An example from Lecture 18: A test for one proportion
H0: p = Ha: p > 0.30
Under H0 assumption
Test statistic:

6. An example from Lecture 18: A test for one proportion
With a p-value smaller than 0.05, we reject the null hypothesis.
This means we have statistically significant evidence that more than 30% of this age group has driven while drowsy in the past year.
P-value = P(Z>1.89) =
0.0294

7. Today: We consider T statistics (for means) instead of Z statistics (for proportions)
The formula is still:
Check: n>30 or normal sample
In the case of one mean, we have:

8. Today: We consider T statistics (for means) instead of Z statistics (for proportions)
The formula is still:
Check: n>30 or normal sample
In the case of the mean of paired differences, we have:

9. Today: We consider T statistics (for means) instead of Z statistics (for proportions)
The formula is still:
Check: n>30 or normal sample fpr each sample
In the case of difference of two means, we have:

10. Motivating example: is my spin class hurting my hearing?
Loud music and spin classes go hand-in-hand, but is the music loud enough to permanently damage hearing?
The threshold for hearing loss is accepted to be 85 db.
Research Question: Does the average music volume at a spin class exceed 85 db?

11. Yes – data looks normalish
Measured variable: the noise level (in decibels) of music played during exercise classes at a local gym
Do the data suggest that the population mean noise level is greater than 85 decibels?
Sample from Survey:
n = 20 classes
mean = 89.8 decibels
s = 7.64 decibels
Can we use the one-sample t procedure?
Yes – data looks normalish
State the hypotheses:
H0: µ = Ha: µ > 85

12. H0: μ = 85 decibels Ha: μ > 85 decibels
Calculate and interpret the t test statistic
H0: μ = 85 decibels Ha: μ > 85 decibels
Sample from Survey:
n = 20 classes; mean = 89.8 decibels; st dev = 7.64 decibels
d.f. = n – 1 = 19
Interpret t statistic: Our sample mean of 89.8 is
2.81 standard deviations above he population mean of 85, assuming the null is true.

13. Find the p-value in Minitab
Select: t table and df = 19
Graph -> Probability
Distribution Plots
select

14. Find the p-value in Minitab (continued)
H0: μ = 85 decibels
Ha: μ > 85 decibels
T = 2.81
creates p-value
picture found on
next slide

15. p-value picture - one-sided test
H0: μ = 85 decibels
Ha: μ > 85 decibels
df = 19
Interpretation: The likelihood of getting a t-statistic at least as _____ as 2.81, assuming the null hypothesis is true, equals
large

16. What can go wrong: Two types of errors
This comes from Section 12.1.
Consider a medical test for a disease in which the hypotheses are
H0: You do not have the disease.
Ha: You do have the disease.
If you decide to reject H0 but you turn out to be incorrect, this is a false positive.
If you decide not to reject H0 but you turn out to be incorrect, this is a false negative.

17. What can go wrong: Two types of errors
This comes from Section 12.1.
Consider a medical test for a disease in which the hypotheses are
H0: You do not have the disease.
Ha: You do have the disease.
A false positive is called a type I error.
A false negative is called a type II error.

18. What can go wrong: Two types of errors
This comes from Section 12.1.
Generally speaking:
• A type I error occurs when you erroneously reject H0.
• A type II error occurs when you erroneously fail to reject H0.
The power of a test is the probability, given a particular alternative is true, of rejecting H0.
The level of significance is the p-value cutoff for rejecting H0. (Often it’s 0.05.) If H0 is true, the level of significance is the probability of a type I error.

19. Decision and conclusion: Clicker Question
H0: μ = Ha: μ > p-value = 0.006
Can reject H0 in favor of Ha. Can claim μ > 85. Type 2 error is now possible.
Can reject H0 in favor of Ha. Can claim μ > 85. Type 1 error is now possible.
Can’t reject H0 in favor of Ha. Cannot claim μ > 85. Type 2 is error is now possible.
Can’t reject H0 in favor of Ha. Cannot claim μ > 85. Type 1 error is now possible.

20. Hypothesis Tests: One Sample
Response
Variable
Categorical
Quantitative
Population
Parameter
Sample
Estimate
Inferential
Procedure
one-proportion Z
one-sample t
Test statistic
Chapter 12 13
P-value
Z table
T table with df = (n – 1) because σ is unknown

21. Which procedure should we use?
What kind of data do we have?
What kind of research question do we have?
Show something specific
Estimate a population parameter
Categorical
(binomial)
Quantitative
Pop. mean:
Hypothesis
Test
Confidence interval
Pop. proportion: p

22. Example:
Researchers at Penn State are interested in estimating the percentage of underage students who drink alcohol at least once a month. They should…
Construct a confidence interval for a population proportion
Perform a hypothesis test for a population proportion
Construct a confidence interval for a population mean
Perform a hypothesis test for a population mean

23. What is the average number of texts that a STAT 200 student receives on a weekday? To answer this question we should…
Construct a confidence interval for a population proportion
Perform a hypothesis test for a population proportion
Construct a confidence interval for a population mean
Perform a hypothesis test for a population mean

24. Do more than 30% of Penn State students binge drink at least once a month on average? To answer this question we should…
Construct a confidence interval for a population proportion
Perform a hypothesis test for a population proportion
Construct a confidence interval for a population mean
Perform a hypothesis test for a population mean

25. If you understand today’s lecture…
13.1, 13.3, 13.17, 13.19, 13.23, 13.25, 13.63, 13.67, 13.99
Objectives:
• Formulate null and alternative hypotheses involving population means
• Calculate T-statistics (for means) instead of Z-statistics (for proportions); determine correct degrees of freedom.
• Determine correct test procedure from context of problem.
• Identify potential errors that can occur in hypothesis testing; distinguish between type I and type II errors.