1. Warm-up
Find the z-score for the 90th percentile in a Normal Distribution:
What is the Law of Large Numbers and the Central Limit Theorem.
Law of Large Numbers- the sample mean (or proportion) from a large SRS will be close to unknown population mean (or proportion)
Central Limit Theorem- When n >30 the sampling distribution of the sample means will be approximately normal
Answer to 1: InvNorm(.90) = 1.28

2. Confidence Intervals
Chapter 10.1

3. Lottery Scandal
For the Vietnam War, the US government used a lottery system to draft soldiers. The lottery chose people by randomly choosing birthdates. Upon analysis, the correlation between birth date and draft number was r = (r should = 0 if random). It was discovered men born later in the year tended to be assigned lower draft numbers. We could calculate that this has probability of less than of occurring in a true random system. Therefore, there is strong evidence that the lottery system was biased.

4. Gender Bias Scandal
An investigation of 20 new businesses started by women and 20 new businesses started by men revealed that after 2 years 12 of those started by women failed while only 8 of those started by men failed. Is this evidence that failures are more frequent among businesses started by women than those started by men? Well, calculations reveal that this could occur by chance in nearly one-fifth of businesses by chance. So, the evidence is not convincing.

5. Hero Probability Expresses the strength of our conclusions
Protects us from jumping to conclusions when only chance variation is at work

6. Estimation
The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.
Ex: the sample mean ( 𝑥 ) is employed to estimate the population mean ( μ ).

7. Estimation There are two types of estimators: Point Estimator
Interval Estimator
An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval.
That is we say (with some ___% certainty) that the population parameter of interest is between some lower and upper bounds.

8. Confidence Interval for μ
Has two parts
Interval: estimate ± margin of error
Confidence Level: gives the percent of time that the interval will capture the true parameter value in repeated samples

9. Completing a Confidence Interval
Must include all of the following:
P: Identify the population of interest
A: Assumptions (cannot find a confidence interval without these)
N: Name the inference procedure you are using
I: find your confidence interval
C: Interpret your result in the context of the problem

10. Assumptions/Conditions
Independence (selecting one element of sample does not influence selection of the next):
Data comes from an SRS from the population of interest and the sample is large enough that population ≥ 10n.
Normality:
Sample came from a population this normally distributed (given or graph) Or
Sample is large enough for CLT
Sampling distribution of sample mean is approximately normal (n ≥30).

11. Confidence Interval

12. Lower Confidence Upper Confidence Limit Limit

13. Confidence Levels What z-score (critical value)?
Use invNorm to find z-score
For a 90% confidence interval you need the critical value for 90%. You have two tails that make up the other 10%. To get z-score for .10/2 = = InvNorm.95 =

14. Goals for Interval Behavior
User choose the confidence level
The margin of error follows
We want a high confidence level
We want small margin of error
To find out the size n you need to get a small margin of error, work backwards.

16. What does it mean to be 95% confident?
The resulting Interval will capture the true mean or it will not, so it is incorrect to say:
There is a 95% chance that the true mean falls in this interval….

17. What does it mean to be 95% confident?
You can say:
You are 95% confident that the true mean is in your interval….meaning that the method produces a confidence interval that captures 𝜇 95% of the time.
Saying you are 95% confident means if you take random samples repeatedly from this population and compute a confidence interval from each sample, in the long run 95% of these intervals would contain or capture 𝜇.

18. Statement: (memorize!!)
We are ________% confident that the true mean context lies within the interval ______ and ______.

19. 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?

20. N: We will be using a 90% confidence interval I:
Answer:
P: We are interested in the mean Potassium level of blood
A: Assume independence since we have a SRS of blood measurements and the population is greater than 10(3).
Potassium level is normally distributed (given)
σ known
N: We will be using a 90% confidence interval I:
C: We are 90% confident that the true mean potassium level is between 3.01 and 3.39.

21. 95% confidence interval?
We are 95% confident that the true mean potassium level is between 2.97 and 3.43.

22. 99% confidence interval?
We are 99% confident that the true mean potassium level is between 2.90 and 3.50.
What happens to interval as confidence level increases?

23. Choosing Sample Size
You can have both high confidence and a small margin of error by taking enough observations.
Example: Company management wants a report of the mean screen tension for the day’s production accurate to within ±5 mV with 95% confidence. How large a sample of video monitors must be measured to comply with this request? (given σ = 43)
For 95% confidence the table gives us a critical value (z-score) of 1.96

24. m ≤ 5
z* 𝜎 𝑛 ≤ 5
𝑛 ≤ 5
43 𝑛 ≤ 2.551
43 ≤ 𝑛
≤ 𝑛
≤ 𝑛
≤ n, so take 285

25. Example:
A manufacturer of high-resolution video terminals must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. Too much tension will tear the mesh and too little will allow wrinkles. The tension is measured by an electrical device with output readings in millivolts (mV). Some variation is inherent in the production process. Careful study has shown that when the process is operating properly, the standard deviation of the tension readings is σ =43 mV. Here are the tension readings from an SRS of 20 screens from a single day’s production.
269.5
297
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310
343.3
328.1
342.6
338.8
340.1
374.6
336.1

26. Construct a 90% confidence interval for the mean tension μ of all the screens produced in this day.
P: The population of interest is all of the video terminals produced on the day in question. We want to estimate μ, the mean tension for all of these screens
A: Assume independence since the data comes from a SRS of 20 screens from the population of all screens produced that day. According to the normal probability plot the sample is approximately normal (also remember if population is approximately normal, sample is too).

27. N: Confidence interval for population mean will be used
I: Compute mean:
𝑥 = mV. Plug numbers into formula:
𝑥 ±z* 𝜎 𝑛
306.3 ±
= (290.5, 322.1)
C: We are 90% confident that the true mean tension in the entire batch of video terminals produced that day is between and mV.