1. Lecture 4 Chapter 11 wrap-up
Chapter Inference about the mean when the s.d. is unknown
Chapter 12.3 – Inference about a population proportion

2. Hypothesis Testing – Basic Steps
Set up alternative and null hypotheses
Calculate test statistic, e.g. z-score
Find critical values and compare the test statistic to critical value (rejection region method) or find p-value (p-value method)
Make substantive conclusions.

3. Right-, Left, Two-Sided Tests
Right-sided:
Left-sided:
2-sided:

4. Summary: Steps in Testing
Determine and (right//left//2sided), and decide on a significance level
Rejection region method: calculate and reject if // //
P-value method: calculate => P(Z>z) // P(Z<z) // P(|Z|>|z|) from z-tables, or “Prob>z” // ”Prob<z” // “Prob>|z|” from JMP; reject if p-value .
Interpret the result and tell a story.

5. Relationship Between CIs and Hypothesis Tests
There is a duality between confidence intervals and hypothesis tests
We can construct a level hypothesis test based on a level confidence interval by rejecting if and only if is not in the confidence interval
We can construct a level confidence interval based on a level hypothesis test by including in the confidence interval if and only if the test does not reject

6. Calculation of Type II error
State alternative for which you want to find P(Type II error).
Find rejection region in terms of unstandardized statistic (sample mean)
Find the probability of the sample mean falling outside the rejection region if the alternative under consideration is true (use standardization relative to the alternative hypothesis mean to calculate this probability).

7. Summary: Power Calculations
Works only for the rejection region method, and we don’t do it for 2-sided tests.
Calculate for level- test.
Right-sided: P(Z<z) from z-table; =P(Z<z)
Left-sided: P(Z>z) from z-table; =P(Z>z)

8. Frequent values
0.10
0.05
0.025
0.01
0.005
1.28
1.64
1.96
2.33
2.58

9. Practice Problems
11.68,11.84,12.40,12.46

10. Chapter 12
In this chapter we utilize the approach developed before to describe a population.
Identify the parameter to be estimated or tested.
Specify the parameter’s estimator and its sampling distribution.
Construct a confidence interval estimator or perform a hypothesis test.

11. 12.2 Inference About a Population Mean When the Population Standard Deviation Is Unknown
Recall that when s is known we use the following
statistic to estimate and test a population mean
When s is unknown, we use its point estimator s, and the z-statistic is replaced then by the t-statistic

12. t-Statistic
When the sampled population is normally distributed, the t statistic is Student t distributed with n-1 degrees of freedom.
Confidence Interval: where is the quantile of the Student t-distribution with n-1 degrees of freedom.

13. t-Statistic
When the sampled population is normally distributed, the t statistic is Student t distributed with n-1 degrees of freedom.
Confidence Interval: where is the quantile of the Student t-distribution with n-1 degrees of freedom.

14. The t - Statistic t s The “degrees of freedom”,
(a function of the sample size)
determine how spread the
distribution is (compared to the
normal distribution)
The t distribution is mound-shaped,
and symmetrical around zero.
d.f. = v2
d.f. = v1
v1 < v2

15. A
= .05 tA
t.100
t.05
t.025
t.01
t.005

16. Testing m when s is unknown
Example 12.1
In order to determine the number of workers required to meet demand, the productivity of newly hired trainees is studied.
It is believed that trainees can process and distribute more than 450 packages per hour within one week of hiring.
Fifty trainees were observed for one hour. In this sample of 50 trainees, the mean number of packages processed is and s=38.82.
Can we conclude that the belief is correct, based on the productivity observation of 50 trainees?

18. Checking the required conditions
In deriving the test and confidence interval, we have made two assumptions: (i) the sample is a random sample from the population; (ii) the distribution of the population is normal.
The t test is robust – the results are still approximately valid as long as the population is not extremely nonnormal. Also if the sample size is large, the results are approximately valid.
A rough graphical approach to examining normality is to look at the sample histogram.

19. JMP Example
Problem 12.45: Companies that sell groceries over the Internet are called e-grocers. Customers enter their orders, pay by credit card, and receive delivery by truck. A potential e-grocer analyzed the market and determined that to be profitable the average order would have to exceed $85. To determine whether an e-grocer would be profitable in one large city, she offered the service and recorded the size of the order for a random sample of customers. Can we infer from the data that e-grocery will be profitable in this city at significance level 0.05?

20. 12.3 Inference About a Population Variance
Sometimes we are interested in making inference about the variability of processes.
Examples:
The consistency of a production process for quality control purposes.
Investors use variance as a measure of risk.
To draw inference about variability, the parameter of interest is s2.

21. 12.3 Inference About a Population Variance
The sample variance s2 is an unbiased, consistent and efficient point estimator for s2.
The statistic has a distribution called Chi-squared, if the population is normally distributed.
d.f. = 5
d.f. = 10

22. Confidence Interval for Population Variance
From the following probability statement P(c21-a/2 < c2 < c2a/2) = 1-a we have (by substituting c2 = [(n - 1)s2]/s2.)

23. Testing the Population Variance
Example 12.3 (operation management application)
A container-filling machine is believed to fill 1 liter containers so consistently, that the variance of the filling will be less than 1 cc (.001 liter).
To test this belief a random sample of 25 1-liter fills was taken, and the results recorded (Xm12-03). s2=
Do these data support the belief that the variance is less than 1cc at 5% significance level?
Find a 99% confidence interval for the variance of fills.

25. JMP implementation of two-sided test

26. 12.4 Inference About a Population Proportion
When the population consists of nominal data (e.g., does the customer prefer Pepsi or Coke), the only inference we can make is about the proportion of occurrence of a certain value.
When there are two categories (success and failure), the parameter p describes the proportion of successes in the population. The probability of obtaining X successes in a random sample of size n from a large population can be calculated using the binomial distribution.

27. 12.4 Inference About a Population Proportion
Statistic and sampling distribution
the statistic used when making inference about p is:
Under certain conditions, [np > 5 and n(1-p) > 5], is approximately normally distributed, with m = p and s2 = p(1 - p)/n.

28. Testing and Estimating the Proportion
Test statistic for p
Interval estimator for p (1-a confidence level)

29. Testing the Proportion
Example 12.5 (Predicting the winner in election day)
Voters are asked by a certain network to participate in an exit poll in order to predict the winner on election day.
The exit poll consists of 765 voters say that they voted for the Republican network.
The polls close at 8:00. Should the network announce at 8:01 that the Republican candidate will win?

31. Selecting the Sample Size to Estimate the Proportion
Recall: The confidence interval for the proportion is
Thus, to estimate the proportion to within W, we can write
The required sample size is:

32. Sample Size to Estimate the Proportion
Example
Suppose we want to estimate the proportion of customers who prefer our company’s brand to within .03 with 95% confidence.
Find the sample size needed.
Solution
W = .03; 1 - a = .95,
therefore a/2 = .025,
so z.025 = 1.96
Since the sample has not yet
been taken, the sample proportion
is still unknown.
We proceed using either one of the
following two methods:

33. Sample Size to Estimate the Proportion
Method 1:
There is no knowledge about the value of
Let This results in the largest possible n needed for a a confidence interval of the form
If the sample proportion does not equal .5, the actual W will be narrower than .03 with the n obtained by the formula below.
Method 2:
There is some idea about what will turn out to be.
Use a probable value of to calculate the sample size

34. Practice Problems
12.40, 12.46, 12.58, 12.77, 12.98