Statistics 1b
Lecture 1, 3-02-2014
Ways of learning about parameters:
1. Confidence intervals (“What values do I think are plausible for µ?”)
2. Statistical significance testing (“Is it plausible that µ=100”)
Facts about statistics we know:
1. If Xbar is a random variable with mean u and standard deviation o and we take a srs of N
observations: has mean u and standard deviation o/√N
2. Central Limit Theorem: As N -> ∞, Xbar becomes distributed more and more normally.
3. Linear functions of normals are normal. Linear combinations of normal are normal.
4. With reasonable sample sizes, we can assume: Xbar~Normal(µ,o√N)
5. X is within a certain distance from µ 95% of the time.
6. From the z-table: 2,5% of area of normal is less than 1.96
7. Xbar is within 1.96 o/√N of µ 95% of the time
8. o/√N is called the standard error of Xbar
9. X is within 1.96 standard errors of µ 95% of the time.
10. This means… µ is within 1.96 standard errrs of X 95% of the time
11. If z* is the z score where alpha/2 of the area of the normal is less than z*, then:
Xbar +z* o√N is a 100(1-alpha)% confidence interval for µ.
alpha (the error rate) determines the “size” of the confidence interval.
12. To build a confidence interval using z scores, we must know:
- Xbar (And that Xbar is approximately normal)
- O (standard deviation)
- N
- The desired alpha for Confidence Interval (CI)
Steps to creating confidence interval
1. Given confidence level, find alpha/2. If no confidence level is given it is 95%!
2. Use z table or software to find z* corresponding alpha/2
3. Compute standard error (SE), o√N
4. Compute CI: Xbar + z*SE
5. Interpret CI
Confidence interval is an interval that assures that if repeated over and over it will give the same value.
We are 95% confident that u is within the confidence interval, because 95% of confidence intervals of this
size will contain u.
The confidence interval is NOT a probability statement.
Formula confidence interval: Xbar ± z* x SE
1
Lecture 1, 3-02-2014
Ways of learning about parameters:
1. Confidence intervals (“What values do I think are plausible for µ?”)
2. Statistical significance testing (“Is it plausible that µ=100”)
Facts about statistics we know:
1. If Xbar is a random variable with mean u and standard deviation o and we take a srs of N
observations: has mean u and standard deviation o/√N
2. Central Limit Theorem: As N -> ∞, Xbar becomes distributed more and more normally.
3. Linear functions of normals are normal. Linear combinations of normal are normal.
4. With reasonable sample sizes, we can assume: Xbar~Normal(µ,o√N)
5. X is within a certain distance from µ 95% of the time.
6. From the z-table: 2,5% of area of normal is less than 1.96
7. Xbar is within 1.96 o/√N of µ 95% of the time
8. o/√N is called the standard error of Xbar
9. X is within 1.96 standard errors of µ 95% of the time.
10. This means… µ is within 1.96 standard errrs of X 95% of the time
11. If z* is the z score where alpha/2 of the area of the normal is less than z*, then:
Xbar +z* o√N is a 100(1-alpha)% confidence interval for µ.
alpha (the error rate) determines the “size” of the confidence interval.
12. To build a confidence interval using z scores, we must know:
- Xbar (And that Xbar is approximately normal)
- O (standard deviation)
- N
- The desired alpha for Confidence Interval (CI)
Steps to creating confidence interval
1. Given confidence level, find alpha/2. If no confidence level is given it is 95%!
2. Use z table or software to find z* corresponding alpha/2
3. Compute standard error (SE), o√N
4. Compute CI: Xbar + z*SE
5. Interpret CI
Confidence interval is an interval that assures that if repeated over and over it will give the same value.
We are 95% confident that u is within the confidence interval, because 95% of confidence intervals of this
size will contain u.
The confidence interval is NOT a probability statement.
Formula confidence interval: Xbar ± z* x SE
1
