1. Estimating with Confidence
Chapter 6.1
Estimating with Confidence

2. Point estimation
Sample mean is the natural estimator of the unknown population mean.
Is the point estimation a good method?
1. It may never hit the true value (population mean).
2. We have no idea about the variability of the estimation. Therefore, we have no confidence about how close our estimator is to the true value.
Net Gun (flyswatter)
Taser Gun
Idea: It is better to use an INTERVAL than a POINT estimator.

3. Review: Chapter 1.3: All Normal curves N(m,s) share 68-95-99.7 Rule
About 68% of all observations are within 1 SD (s) of mean (m).
Called: C=68%, z*≈1
About 95% of all observations are within 2 s of the mean m.
Called: C=95%, z* ≈ 2
Almost all (99.7%) observations are within 3 s of the mean.
Called: C=99.7%, z* ≈ 3
Going to an example from the book on women’s heights, the mean here was 64.5, standard deviation 2.5 inches.
When we talk about the mean and standard deviation with respect to the curve instead of the actual sample, we use different notation.
Mu for mean, sigma for sd. If you consider the area under the curve to represent all of the individuals, then you can divide it into chunks to represent parts of the whole. Like if you divided it down the middle, half of the people are in each half. Here it is divided up into parts not through the middle but by lines that are 1, 2 or 3 standard deviations away from the mean. If you look at the center, pink part, it is the area 1 sd on either side of the mean. By definition for normal curves, this area is 68% of the total. So if you know the mean and sd, you also know that 68% of women are between 62 and 67 inches tall. Similarly for the areas defined by lines drawn 2 or 3 sd from the mean. We might want to know what percent of women are over 72 inches tall. That is 3 sd. We can see that 99.7 percent of women are less than 72 or greater than 57. Or that .3 percent of women are really tall or really short. Since the distribution is symmetric, we can divide by two to find the percent of women that are really tall: .15%
You need to be able to work problems like I just did - bunch in book. But what if you want to know something not defined by the sd? Like, what percentage of women are taller than 68 inches? Know that half are smaller than And that half of this middle area, 34%, are smaller than 67 inches, so = 84% are smaller than 67, or 16% are larger than 67 inches. But you want to know the proportion larger than 68 inches. You can look this up on a table, but first you have to do something called standardizing. The reason is that although all normal curves share the properties shown above, they differ by their mean and standard deviation. You would have to have a different table for every curve. When you standardize a normal distribution, you change it so the mean is 0 and the sd is 1. Any normal distribution can be standardized.
Standard Normal Distribution N(0, 1)
Reminder: µ (mu) is the mean of the idealized curve, while x¯ is the mean of a sample.
s (sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample.

4. Confidence levels
Confidence intervals contain the population mean m in C% of samples. Different areas under the curve give different confidence levels C.
z*:
z* is related to the chosen confidence level C.
C is the area under the standard normal curve between −z* and z*. C
The confidence interval is thus:
−z* z*
Example: For an 80% confidence level C, 80% of the normal curve’s area is contained in the interval.

5. Point estimation versus interval
When population mean (µ) is unknown, it is better to use an interval than a point to estimate it.
The theory behind interval estimation looks at the sampling distribution of the statistic.
Confidence level C- CI for the population mean µ is :
For a particular confidence level, C, the appropriate z* value is given in the last row of Table D.
Example: For a 98% confidence level, z*=2.326

6. 99% CI for the population mean µ is : i.e.: C=99%, z*=2.576
Specific Confidence Intervals for population mean
99% CI for the population mean µ is :
i.e.: C=99%, z*=2.576
95% CI for the population mean µ is :
i.e.: C=95%, z*=1.960
90% CI for the population mean µ is :
i.e.: C=90%, z*=1.645

7. Link between confidence level and margin of error
The margin of error depends on z.
Higher confidence C implies a larger margin of error m (thus less precision in our estimates).
A lower confidence level C produces a smaller margin of error m (thus better precision in our estimates). C z*
−z*
m m

8. Example 1
The average lifetime of 36 randomly selected certain brand TVs is 20 years. Suppose the SD of all TVs is 2 years.
Construct a 95% CI for the average lifetime of all TVs from this brand.
A 95% CI for the average lifetime of all TVs from this brand is:
(19.35, 20.65)

9. Example 2
1. The average height of 100 randomly selected UNCW students is 5.9 feet. Suppose the SD of the heights of all students is 1.2 feet.
Construct 99%, 95% and 90% CIs for the average height of all students.
A 99% CI for the average height of all students is:
(5.5904, )
A 95% CI for the average height of all students is:
(5.6648, )
A 90% CI for the average height of all students is:
(5.702, 6.098)
Note: Confidence level C gets smaller, CI gets smaller

10. Example 2 (Continue)
1. The average height of 100 randomly selected UNCW students is 5.9 feet. Suppose the SD of the heights of all students is 1.2 feet.
Find MOE and construct a 95% CI for average height of all students.
Note: Confidence level C gets smaller, CI gets smaller
A 95% CI for the average height of all students is:
(5.6648, )
2. (Continue…) Select another set of 100 UNCW students randomly. The average height of second set of 100 students is 5.5 feet. Suppose the SD of the heights of all students is 1.2 feet.
Find MOE and construct 95% CIs for average height of all students.
A 95% CI for the average height of all students is:
(5.2648, )

11. Outlines for Z* Z* depends on the level of confidence C.
What does “confidence” mean? This idea is only true for simple random samples and completely randomized experiments.
Margin of error: Z*/√(n)
See applet: In Statcrunch, use Applet option and select desired options. Then run the applet.

12. Understanding of Confidence Intervals
With 95% confidence, we can say that µ should be within roughly 2 standard deviations (that is, 2*s/√n) from our sample mean .
About 95% of all possible samples of this size n, µ will indeed fall in our confidence interval.
About only 5% of samples would be farther from µ.
applet.

13. Example 3
A 90% CI for the average life that this medicine can prolong for all cancer patients is: (3.8453, );
Z*=1.645; MOE=(1.645)*(0.75)/sqrt(n)=0.1; so n=(1.645*0.75/0.1)^2=
We will take n=153.

14. Summary to Confidence Interval
If Confidence level C gets larger and n stays the same, what will happen to z*, MOE, CI, and prediction precision?
If Z* and  stay the same, when n goes bigger, what will happen to MOE and CI?