Society Investigating Mathematical Mind-Expanding Recreations
Trial by Jury | Statistical Hypothesis Testing |
Prosecutor | Statistician |
Trial | Collection of Data |
Jury decides on the verdict | Statistical test |
Assume defendant is innocent | Assume the null hypothesis is true |
Weigh the evidence provided by | Assess the evidence provided by |
testimony and exhibits | the data (as summarized in the test statistic) |
assuming defendant is innocent | assuming null hypothesis is true |
Evidence against the defendant | Calculate a p-value for the test statistic |
assuming defendant is innocent | assuming null hypothesis is true |
Defendant found guilty | Reject the null hypothesis if |
beyond a reasonable doubt | p-value less than the significance level |
The Law of Large Numbers states that as the sample size (number of
observations) increases, the sample mean will approach the actual mean.
the Central Limit Theorem states that as the sample size increases, the distribution of becomes closer to a normal distribution. Also, the distribution of the sum of the random observations, becomes closer to a normal distribution.
Preferred Newspaper | |||
Gender | Globe and Mail | Toronto Star | Toronto Sun |
Male | |||
Female | |||
If gender and newspaper are truly independent, has a distribution on
rc-1-(r-1)-(c-1)=(r-1)(c-1)degrees of freedom, where r is the number of rows in our table and c is the number of columns.
Note 1: We lose a degree of freedom each time we treat something as fixed, for example, the total number of males, the total number of Sun readers, etc.
Note 2: The distribution of X^2 follows from the above distribution theory, plus some calculation. See, for example, Mathematical Statistics with Applications, by Mendenhall, Wackerly, and Scheaffer.
The distribution of the test statistic assuming the null hypothesis
is true: with (r-1)(c-1) degrees of freedom.
The conclusion: If the probability of getting
an that is as large or larger
than what we got is small, we have evidence that our null
hypothesis is false.
www.dartmouth.edu/~chance/
www.math.montana.edu/mathed/simms/
Newspaper | |||
Gender | Globe | Star | Sun |
Male | |||
Female |
Words | Sense and Sensibility | Emma | Sanditon I | Sandition II |
a PB such | 14 | 16 | 8 | 2 |
a NPB such | 133 | 180 | 93 | 81 |
and FB I | 12 | 14 | 12 | 1 |
and NFP I | 241 | 285 | 139 | 153 |
the PB on | 11 | 6 | 8 | 17 |
the NPB on | 259 | 265 | 221 | 204 |
Hit | No hit | |
Regular season | 2584 | 7280 |
World Series | 35 | 63 |
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