Confidence Intervals and Hypotesis Testing

Question 1

What is a confidence interval?

What are the 3 elements needed to create one?

So we can say that confidence interval = …

Answer 1

• a range of values within which we expect the true population parameter to lie with a certain lvl of confidence.

1. a sample estimate of the pop. parameter;
2. a margin of error MOE;
3. a confidence level alfa.

C.I. = Sample estimate +- Margin of Error

Question 2

What does the confidence level represents?

What does its complement (1-alfa) represents?

Answer 2

• is not the probability that one specific interval contains the population parameter. If we were to take many random samples, the confidence level tells us how many confidence intervals, out of all the calculated confidence intervals, would contain the true parameter of the population.
• its complement represents the significance level: out of all the calculated confidence intervals, how many don’t contain the true parameter of the pop

Question 3

Margin of error MOE:
- how is it calculated? MOE = … x …
- what does the first … represent? the second …?
- How does the confidence level influence MOE?

Answer 3

• MOE = standard error x reliability factor;
• standard error is the stddev of the sample estimate;
• the reliability factor is a value that depends on the chosen confidence level.

The bigger the confidence lvl, the bigger the MOE.

Question 4

Sample Mean:
- if we know the variance of the pop than the confidence interval CI = … x …
-if we don’t know the variance of the pop than the confidence interval CI = … x …

Sample proportion: CI = … x …

Answer 4

• CI = xtrattino +- z alfa/2 * stddev o/sqrt(n);
• CI = xtrattino +- t alfa/2 * stddev s/sqrt(n);
• CI = pcappello +- z alfa/2 * sqrt(pq/n)

Question 5

What is a hypothesis?

What does the null hypothesis Ho represent? What do we have to decide? What does it contain?

What does the alternative hypothesis H1 represent?

Answer 5

• a statement about a parameter of the population.
• It is the status quo: what is assumed to be true until evidence suggests otherwise.
  We have to decide if Ho can be rejected or not.
  Contains =, >=, <=.
• It challenges the status quo H0, it’s what we wwant to demonstrate.
  Contains !=, <, >.

Question 6

P-Value:
- what probability is it?
- how do you calculate it?

Answer 6

• the probability, assuming H0 is true, that the test statistic takes a value as or more extreme than the value observed.
  P(Z > Zobs |mu=muo)

1. calculate the observed test statistic
2. calculate the z/t score

Question 7

Type 1 and 2 errors:
- example scenario, H0, H1.
- what 2 possibilities do we have in a test?

Type 1 Error:
- when does it happen?
- example
- what is the significance level?

Type 2 error:
- when does it happen?
-example

Answer 7

• scenario: we are testing patients for disease. H0 (no disease), H1(yes disease).
• we can reject H0 or don’t reject H0.

Type I:
-when we wrongly reject H0 even if it is actually true.
- positive test even if no disease (false positive).
- the risk of making a type I error.

Type II:
- when we wrongly don’t reject H0 even if it is actually false.
- negative test even if yes disease (false negative).

Question 8

Hypothesis tests:
- 3 types
- 2 possibilities

Answer 8

Two-tailed:
H0: mu = muo
H1: mu != muo
1. reject H0 if xtrattino not inside muo+-z alfa/2 * stddev o / sqrt(n).
2. reject H0 if p value < alfa.

Right-tailed:
H0: mu <= muo
H1: mu > muo
1. reject H0 if xtrattino > muo+z alfa * stddev o / sqrt(n).
2. reject H0 if p value < alfa.

Left-tailed:
H0: mu >= muo
H1: mu < muo
1. reject H0 if xtrattino < muo-z alfa * stddev o / sqrt(n).
2. reject H0 if p value < alfa.

Question 9

At the end of an hypothesi test we can say… 2 things

Answer 9

• there is not enough statistical evidence to reject H0.
• there is enough statistical evidence to reject H0.