1. Confidence Intervals
Chapter 8

2. Rate your confidence 0 - 100 Name my age within 10 years?
Shooting a basketball at a wading pool, will make basket?
Shooting the ball at a large trash can, will make basket?
Shooting the ball at a carnival, will make basket?

3. What happens to your confidence as the interval gets smaller?
The larger your confidence, the wider the interval.
Simulation

4. Point Estimate
Use a single statistic based on sample data to estimate a population parameter
Simplest approach
But not always very precise due to variation in the sampling distribution

5. estimate + margin of error
Confidence intervals
Are used to estimate the unknown population mean
Formula:
estimate + margin of error

6. Margin of error Shows how accurate we believe our estimate is
The smaller the margin of error, the more precise our estimate of the true parameter
Formula:

7. Confidence level
Is the success rate of the method used to construct the interval
Using this method, ____% of the time the intervals constructed will contain the true population parameter

8. Critical value (z*) z*=1.645 z*=1.96 z*=2.576 .05 .025 .005 90% 95%
Found from the confidence level
The upper z-score with probability p lying to its right under the standard normal curve
Confidence level tail area z*
.05
.025
z*=1.645
.005
z*=1.96
z*=2.576
90%
95%
99%

9. What does it mean to be 95% confident?
The probability that the interval contains m is 95%
The method used to construct the interval will produce intervals that contain m 95% of the time.

10. Confidence interval for a population mean:
Standard deviation of the statistic
Critical value
estimate
Margin of error

11. Steps for doing a confidence interval:
PANIC!!!!! / State, Plan, Do, Conclude
P-arameter: are we estimating something about means, proportions, slopes?
A-ssumptions
SRS from population
Sampling distribution is normal (or approximately normal)
Given (normal)
Large sample size (approximately normal)
Graph data (approximately normal)
N-ame Interval
I-nterval: Show work
C-onclusion: Write a statement about the interval in the context of the problem.

12. Statement: (memorize!!)
We are ________% confident that the interval ______ to ______ captures the true parameter context.

13. A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with s = A random sample of three has a mean of What is a 90% confidence interval for the mean potassium level?
Parameter: true mean level of potassium in the blood
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed (given)
s known
Name the interval: one-sample z-interval
Interval:
Conclusion:
We are 90% confident that the interval 3.01 to 3.39 captures the true mean potassium level.

14. 95% confidence interval?
Parameter, Assumptions, Interval is the same as previous
Wider Interval due to larger confidence
Interval:
Conclusion: We are 95% confident that the interval from to captures the true mean potassium level.

15. 99% confidence interval?
We are 99% confident that the interval from to captures the true mean potassium level.

16. the interval gets wider as the confidence level increases
What happens to the interval as the confidence level increases?
the interval gets wider as the confidence level increases

17. Homework
Pg ; pg , 29, 31, 33, 34

18. Critical value (z*) z*=1.645 z*=1.96 z*=2.576 .05 .025 .005 90% 95%
Found from the confidence level
The upper z-score with probability p lying to its right under the standard normal curve
Confidence level tail area z*
.05
.025
z*=1.645
.005
z*=1.96
z*=2.576
90%
95%
99%

19. How can you make the margin of error smaller?
z* smaller
(lower confidence level)
s smaller
(less variation in the population)
n larger
(to cut the margin of error in half, n must be 4 times as big)
Really cannot change!

20. A random sample of 50 BGHS students was taken and their mean SAT score was (Assume s = 105) What is a 95% confidence interval for the mean SAT scores of BGHS students?
We are 95% confident that the true mean SAT score for BGHS students is between and

21. Find a sample size: Always round up to the nearest person!
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!

22. The heights of BGHS male students is normally distributed with s = 2
The heights of BGHS 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
Homework pg , 2, 9-19 odd, 25, 26

23. 8.3 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

24. Graph examples of t- curves vs 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

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

26. How to find t* 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

27. Formula: Margin of error Standard deviation of statistic
Critical value
estimate
Margin of error

28. Assumptions for t-inference
Have an SRS from population
s unknown
Normal distribution
Given
Large sample size
Check graph of data

29. Robust
An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated.
t-procedures can be used with some skewness, as long as there are no outliers.
Larger n can have more skewness.

30. Outliers are always a concern, but they are even more of a concern for confidence intervals using the t-distribution
Sample mean is not resistant; hence the sample mean is larger or smaller (drawn toward the outlier) (small numbers of n in t-distribution!)
Sample standard deviation is not resistant; hence the sample standard deviation is larger
Confidence intervals are much wider with an outlier included
Options:
Make sure data is not a typo (data entry error)
Increase sample size beyond 30 observations

31. 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 rates of adults is between and beat per minute.

32. 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.

33. 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 is between and calories.

34. 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.

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

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

37. Homework:
Pg , 71, 73-78