1. Estimating with Confidence: Means and Proportions

2. Today, we will
Learn to get better estimates with our confidence interval for means (using the t distribution)
Learn to generate confidence intervals for proportions

3. REVIEW CIs
We have a sample mean but want to know where the population mean is. To answer this question “what is the population mean?” we construct confidence limits around the sample mean.
95% CI =
where =
and z* is the critical value in a normal probability distribution for computing the upper and lower estimates. Number of standard deviations we need to capture a certain percent of cases (usually 95%)

4. Can we do better?
We can get better estimates under some circumstances!
z distribution (standard normal) is problematic when our sample size is small & sampling distribution is non-normal
We need some way to have our distribution change shape to reflect the uncertainty we have as our sample size changes
That distribution is the t distribution

5. Student’s t Distribution
It is a standardized probability distribution
Unlike z, t changes peakedness as the sample size varies, becoming bell shaped as the sample size increases.
t test for computing confidence limits is
95% CI = with a t for certain degrees of freedom
Degrees of Freedom (df) control the size of the peak based on the sample size
df= n-1

7. t-table
95%
d.f. = (n-1)
90%
98%
80%
99%

8. Steps for C.I. using t Obtain the Std. Error
Get a value of t from the t distribution
Compute the Interval (Plug and Chug)

9. Let’s Practice
Suppose we were interested in how frequently people vote. To study this, a researcher asks 10 people how many times they have voted in the last 5 Congressional Elections
The average number of times a person in this sample voted was 2.7, with a standard deviation of 1.3

10. Step 1: Obtain the Std. Error

11. Step 2: Get a value of t from the t distribution
D.F.= n-1 = 9
Choose a level of risk (.05)
t critical value = 2.262

12. Step 3: Compute the Interval
In repeated samples of the same size from the same population, 95% of samples would yield an interval that contains the true mean.

13. Now You Try
A man drives 30 miles to work every day. There are many stop lights on the way, so it seems to take a different length of time each day. He wants to estimate the average drive time. He times his drive over 25 days and finds a mean drive time of 38 minutes with a standard deviation of 9 minutes.
Using 95% level of confidence, estimate the average drive time with a confidence interval.

14. Step 1: Obtain the Std. Error
Step 2: Get a value of t from the t table
Step 3: Calculate the Confidence interval
df = 24
α = .05
t = 2.064
*1.84
We are 95% confident that the
True mean is between & 41.79

15. Should I use z or t ?
With t, you get more accurate results for smaller sample sizes.
As the degrees of freedom get larger and larger, the t distribution turns into the standard normal distribution (the z distribution)
As a result, we should always use t.
Why did I have to learn this stupid z thing?

16. What if I don’t have a mean
Percentage of people who vote for Bush
Proportion of the population who is in a certain category
We need another test

17. Count = f / n =proportion 100/600 = .60
Surveys and experiments often produce counts which we can turn into proportions.
Count = f / n =proportion
100/600 = .60
Or multiply by 100 to get a percentage
60*100=60%

18. Sampling Error for Proportions
Proportion in a sample is not the same as the True population proportion.
We can estimate Confidence Intervals for proportions just like means

19. The Formulas CI for Means CI = CI for Proportions CI = The Difference

20. p = proportion 1- p = Not p, sometimes called q
Then proportion of people favoring abortion is “p”
The number of people opposing abortion is 1 – p.
If the sample size (n) is “large enough” the sampling distribution will be normal. The sampling distribution from which you are drawing your one sample will approximate a normal probability distribution – a Z distribution when :
N*p > 10 and n(1 – P) > 10
If n*p<10 or n*(1-P)<10, then we must use something else. We will not encounter that this semester

21. Steps for Computing a Confidence Interval for Proportions
Convert the frequency count in your sample into proportions. P = count / sample size, [f/n] /
Find the appropriate critical value of z.
Use the last line on the t table for infinite degrees of freedom (90% 1.645, 95% 1.96, etc…)

22. 3. Calculate the Standard Error
4. Plug and Chug

23. Practice Problem:
Would a majority of all park visitors favor stricter controls on animals (requiring a leash)? Can you be 95% confident that more than half the visitors would approve stricter limits.
Results of the survey were 89 of 150 visitors favored stricter restrictions.
Step 1: Generate Proportion
89/150 = P = or 59.3%

24. Step 3: Compute the Standard Error
Step 2: Find the Critical Value of Z. (Z table or bottom line of t table). This works out to be 1.96
Step 3: Compute the Standard Error
Step 4: Plug and Chug
= *.040 =
Interval is .515 to .671
95% confident that most visitors favor restrictions

25. A National SRS poll of n=500 finds that 330 in the sample favor stronger gun controls. Stated in percent, 66% of this sample favors stronger gun controls.
Step 1: What is the problem? What is the percent of people in the population favoring gun control. Convert the frequency into a proportion 330/500=.66

26. Step 3: Compute the Std. Err.
Step 2: Find the critical Value of Z . If we choose 95% confidence, we use 1.96
Step 3: Compute the Std. Err.
Step 4: Plug and Chug
95% CI = *(.02) =
The mean support for gun control is 66% % =
= .02

27. As the SRS becomes bigger the estimated error around the measure of central tendency gets smaller. The larger the sample the less the chance of getting an atypical average.

28. Choosing between t and z for Confidence Intervals
Proportions: Always use z
Means:
Always use t
Why? If the sample size is large enough to use z, then the t table will give you the right value anyways.