1. Confidence Intervals with Means

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. Guess the number
Teacher will have pre-entered a number into the memory of the calculator.
Then, using the random number generator from a normal distribution, a sample mean will be generated.
Can you determine the true number?

5. 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

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

7. 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:

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

9. 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.

10. Critical value (z*) 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%

11. Confidence interval for a population mean:
Critical value
Standard Error (SD of Sampling Distribution)
estimate
Margin of error

12. CI of Means - Steps: Assumptions – SRS from population
Sample is independent – smaller than 10% of population
Sampling distribution is normal (or approximately normal)
Given (normal)
Large sample size (approximately normal)
Graph data (approximately normal)
σ is known
Calculate the interval

13. 3) Write a statement about the interval in the context of the problem.
Statement: (memorize!!)
We are __________% confident that the true mean context lies within the interval _______ and ______.

14. Confidence Interval Applet
The purpose of this applet is to understand how the intervals move but the population mean doesn’t.

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 σ = 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
Assume patient has time to regenerate blood for each sample so each sample is independent
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. 95% confidence interval?
Assumptions:
Assume same as previous
We are 95% confident that the true mean potassium level is between 2.97 and 3.43.

17. 99% confidence interval? Assumptions: Assume same as previous
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)
σ 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 CHS students was taken and their mean SAT score was (Assume σ = 105) What is a 95% confidence interval for the mean SAT scores of CHS students?
We are 95% confident that the true mean SAT score for CHS 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 CHS 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 CHS male students is normally distributed with σ = 2
The heights of CHS male students is normally distributed with σ = 2.5 inches. How large a sample is necessary to be accurate within +/- .75 inches with a 95% confidence interval?
n = 43