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Statistical Inference and Hypothesis Testing
Dennis Sun
Data 301
Statistical Inference
probability
Population /
Box
Sample /
Data
statistics
The goal of statistics is to infer the unknown population from the
sample.
Probability vs. Statistics
Probability: I have a fair coin. If I toss it 100 times, how many
heads will I get?
0 1
100 draws with
replacement
???
Statistics: I have a coin. I do not know if it is fair or not. I toss it 100
times and get 60 heads. Is the coin fair or not?
? ... ?
100 draws with
replacement
0 , 1 , 1 , ..., 0
|
{z
}
60 1 s
Hypothesis Testing
One method of statistical inference is hypothesis testing.
Idea: Assume a box model. We can see whether the data that was
observed is consistent with that box.
Example: (continued from previous slide) We don’t know whether
the coin is fair or not. But let’s assume that it is. Then the data
should be like 100 draws with replacement from the box 0 1 .
Here’s the distribution of the number of 1 s from that box:
Hypothesis Testing
One method of statistical inference is hypothesis testing.
Idea: Assume a box model. We can see whether the data that was
observed is consistent with that box.
Example: (continued from previous slide) We don’t know whether
the coin is fair or not. But let’s assume that it is. Then the data
should be like 100 draws with replacement from the box 0 1 .
Here’s the distribution of the number of 1 s from that box:
If the coin really were fair,
we would expect to see 60
heads or more just 2.8% of
the time.
The Logic of Hypothesis Testing
We observed 60 heads in 100 tosses. Is the coin fair or not?
1
Assume that it is fair. (This is called the null hypothesis.)
2
If it is fair, then there is just a 2.8% chance that we observe 60
heads or more. (This probability is called the p-value.)
We now have a choice:
3
• Believe the null hypothesis and accept that a 1-in-35 event has
just occurred.
• Reject the null hypothesis and conclude that something else is
going on.
The smaller the p-value, the more unlikely the event you have to
accept to believe the null hypothesis.