1. Confidence Intervals
Chapter 9

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.

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. What does it mean to be 95% confident?
95% chance that m is contained in the confidence interval
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.

9. Critical value (z*) z*=1.645 z*=1.96 z*=2.576 .05 .025 .005
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
z*=1.645
.025
.005
z*=1.96
z*=2.576
90%
95%
99%

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

11. Activity

12. Steps for doing a z-interval for means:
Assumptions –
SRS from population
Sample is < 10% of the population
Independence among data values is plausible
Sampling distribution is normal (or approximately normal)
Given (normal)
Large sample size (n>30)
Graph data (unimodal and relatively symmetric)
s is known
Calculate the interval
Write a conclusion about the interval in the context of the problem.

13. Conclusion:
We are ________% confident that the true mean context lies within the interval ______ and ______.

14. The NAEP (National Assessment of Educational Progress) includes a short test of quantitative skills, covering basic arithmetic and the ability to apply it. The standard deviation of the test is 60. Suppose a random sample of 50 young adult men are taken from a large population. If the sample mean of their scores is 265, what is a 95% confidence interval for the true mean score for young adult men on this test?
What about a 90% confidence interval?

15. 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?
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed (given)
s known
We are 90% confident that the true mean potassium level is between 3.01 and 3.39.

16. Have an SRS of blood measurements
95% confidence interval?
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed (given)
s known
We are 95% confident that the true mean potassium level is between 2.97 and 3.43.

17. 99% confidence interval?
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed (given)
s known
We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

18. 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

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

21. Suppose that we have this random sample of SAT scores:
What is a 95% confidence interval for the true mean SAT score? (Assume s = 105)
We are 95% confident that the true mean SAT score for PWSH students is between and

22. 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!

23. The heights of PWSH male students is normally distributed with s = 2
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

24. In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes 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?

25. 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

26. 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

27. 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

28. How to find t* Use Table 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 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

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

30. Assumptions for t-inference
s unknown
Have an SRS from population
Sample is < 10% of the population
Independence among data values is plausible
Sampling distribution is normal (or approximately normal.
Given (population normal)
Graph data (unimodal and relatively symmetric with no outliers) or large sample size

31. For the Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.
Assumptions:
Have an SRS of healthy, white males
27 white males (placebo group) is <10% of white males
We assume blood pressures are independent
Systolic blood pressure is normally distributed (given).
s is unknown, so we will construct a t-interval
We are 95% confident that the true mean systolic blood pressure of healthy white males is between and

32. 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.

33. Ex. 5 – 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.
(70.883, )

34. 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.

35. Ex. 6 – 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.
(126.16, )

36. A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate?
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.

37. 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

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