1. Confidence Intervals with Means

2. What is the purpose of a confidence interval?
To estimate an unknown population parameter

3. Formula: Margin of error Standard deviation of statistic
Critical value
statistic
Margin of error

4. In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, taking calcium or placebo. The paper reports a mean seated systolic blood pressure of with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed.
Can you find a z-interval for this problem? Why or why not?

5. Student’s t- distribution
Developed by William Gosset
Continuous distribution
Unimodal, symmetrical, bell-shaped density curve
Above the horizontal axis
Area under the curve equals 1
Based on degrees of freedom
df = n - 1

6. t- curves vs standard normal curve
Graph examples of
t- curves vs standard normal curve
Y1: normalpdf(x)
Y2: tpdf(x,2)
Y3:tpdf(x,5) use the -0
Change Y3:tpdf(x,30)
Window: x = [-4,4] scl =1
Y=[0,.5] scl =1

7. How does the t-distributions compare to the standard normal distribution?
Shorter & more spread out
More area under the tails
As n increases, t-distributions become more like a standard normal distribution

8. Standard error – when you substitute s for s.
Formula:
Standard deviation of statistic
Standard error – when you substitute s for s.
Critical value
statistic
Margin of error

9. How to find t* Use Table B for t distributions
Can also use invT on the calculator!
Need upper t* value with 5% is above – so 95% is below
invT(p,df)
Use Table B for t distributions
Look up confidence level at bottom & df on the sides
df = n – 1
Find these t*
90% confidence when n = 5
95% confidence when n = 15
t* = 2.132
t* = 2.145

10. Steps for doing a confidence interval:
Assumptions –
Calculate the interval
Write a statement about the interval in the context of the problem.

11. Statement: (memorize!!)
We are ________% confident that the true mean context is between ______ and ______.

12. Assumptions for t-inference
Have an SRS from population (or randomly assigned treatments)
s unknown
Normal (or approx. normal) distribution
Given
Large sample size
Check graph of data
Use only one of these methods to check normality

13. Ex. 1) Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.
Assumptions:
Have randomly assigned males to treatment
Systolic blood pressure is normally distributed (given).
s is unknown
We are 95% confident that the true mean systolic blood pressure is between and

14. Ex. 2) A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults.
We are 95% confident that the true mean pulse rate of adults is between &

15. Ex 2 continued) Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain.
The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

16. Ex. 3) Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving:
Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt.
We are 98% confident that the true mean calorie content per serving of vanilla yogurt is between calories & calories.

17. Ex 3 continued) A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate?
Note: confidence intervals tell us if something is NOT EQUAL – never less or greater than!
Since 120 calories is not contained within the 98% confidence interval, the evidence suggest that the average calories per serving does not equal 120 calories.

18. Robust
CI & p-values deal with area in the tails – is the area changed greatly when there is skewness
An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the normality assumption is violated.
t-procedures can be used with some skewness, as long as there are no outliers.
Larger n can have more skewness.
Since there is more area in the tails in t-distributions, then, if a distribution has some skewness, the tail area is not greatly affected.

19. Find a sample size:
If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use:
Always round up to the nearest person!

20. Ex 4) The heights of PWSH male students is normally distributed with s = 2.5 inches. How large a sample is necessary to be accurate within inches with a 95% confidence interval?
n = 43

21. Some Cautions:
The data MUST be a SRS from the population (or randomly assigned treatment)
The formula is not correct for more complex sampling designs, i.e., stratified, etc.
No way to correct for bias in data

22. Cautions continued:
Outliers can have a large effect on confidence interval
Must know s to do a z-interval – which is unrealistic in practice