diff --git a/source/ch-Confidence-Intervals.ptx b/source/ch-Confidence-Intervals.ptx index 009ea2c..5e437ca 100644 --- a/source/ch-Confidence-Intervals.ptx +++ b/source/ch-Confidence-Intervals.ptx @@ -9,182 +9,9 @@

-
- Central Limit Theorem + + + -

- Text of section. -

- - - - -

- Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times. - Let S be the number of heads. - Estimate the probability that 34 \leq S \leq 44. -

- - - - - -

- Suppose a fair die is rolled 100 times, and let m be the average value of the rolls. - Estimate the probability that 3.45 \leq m \leq 3.55. -

- - - - - -

- The heights of men in the US have a mean of 69 in and a variance of about 9 in^2, and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in^2. -

- - - - - -

- Suppose the heights of 30 men are sampled, and a sample mean m is taken. - Find the expected value and variance of m. -

- - - - - - -

- Estimate the probability that m \geq 69.5. -

- - - - - - -

- What if the sampled group was women? -

- - - - -
- -
- Confidence Intervals - -

- Text of section. -

- - - - -

- Suppose we flip a coin 100 times and count 60 heads. - Let p be the (unknown) probability that the coin comes up heads on a flip. - Give an approximate 95% confidence interval for the value of p. -

- - - - - -

- Suppose in a sample of 100 people, 12 are left-handed. -

- - - - - -

- Give a 95% confidence interval for the proportion p of left-handed people. -

- - - - - - -

- Give a 90% confidence interval. -

- - - - - - -

- The weights of five mice are measured and recorded below. - Give a 95% confidence interval for the sample mean weight of mice. - (Pretend 5 measurements is large enough for the CLT to apply.) -

- - - - - - - mouse i - 1 - 2 - 3 - 4 - 5 - - - - weight \widetilde{w}_i (g) - 26 - 32 - 33 - 20 - 29 - - -
- - - - - -

- The heights of five plants are measured and recorded below. - Give a 95% confidence interval around the sample mean for the heights of the plants. - (Pretend 5 measurements is large enough for the CLT to apply.) -

- - - - - - - plant i - 1 - 2 - 3 - 4 - 5 - - - - height \widetilde{h}_i (in) - 15 - 14 - 18 - 21 - 17 - - -
- - - -
+ \ No newline at end of file diff --git a/source/docinfo.ptx b/source/docinfo.ptx index cc63dd0..500a474 100644 --- a/source/docinfo.ptx +++ b/source/docinfo.ptx @@ -36,6 +36,8 @@ \DeclareMathOperator{\E}{E} \DeclareMathOperator{\Var}{Var} \DeclareMathOperator{\Cov}{Cov} + + \newcommand{\est}{\widehat} diff --git a/source/sec-CLT.ptx b/source/sec-CLT.ptx new file mode 100644 index 0000000..fa42cb5 --- /dev/null +++ b/source/sec-CLT.ptx @@ -0,0 +1,64 @@ +
+ Central Limit Theorem + +

+ Text of section. +

+ + + + +

+ Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times. + Let S be the number of heads. + Estimate the probability that 34 \leq S \leq 44. +

+ + + + + +

+ Suppose a fair die is rolled 100 times, and let m be the average value of the rolls. + Estimate the probability that 3.45 \leq m \leq 3.55. +

+ + + + + +

+ The heights of men in the US have a mean of 69 in and a variance of about 9 in^2, and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in^2. +

+ + + + + +

+ Suppose the heights of 30 men are sampled, and a sample mean m is taken. + Find the expected value and variance of m. +

+ + + + + + +

+ Estimate the probability that m \geq 69.5. +

+ + + + + + +

+ What if the sampled group was women? +

+ + + + +
\ No newline at end of file diff --git a/source/sec-Confidence-Intervals.ptx b/source/sec-Confidence-Intervals.ptx new file mode 100644 index 0000000..2e85b13 --- /dev/null +++ b/source/sec-Confidence-Intervals.ptx @@ -0,0 +1,113 @@ +
+ Confidence Intervals + +

+ Text of section. +

+ + + + +

+ Suppose we flip a coin 100 times and count 60 heads. + Let p be the (unknown) probability that the coin comes up heads on a flip. + Give an approximate 95% confidence interval for the value of p. +

+ + + + + +

+ Suppose in a sample of 100 people, 12 are left-handed. +

+ + + + + +

+ Give a 95% confidence interval for the proportion p of left-handed people. +

+ + + + + + +

+ Give a 90% confidence interval. +

+ + + + + + +

+ The weights of five mice are measured and recorded below. + Give a 95% confidence interval for the sample mean weight of mice. + (Pretend 5 measurements is large enough for the CLT to apply.) +

+ + + + + + + mouse i + 1 + 2 + 3 + 4 + 5 + + + + weight \widetilde{w}_i (g) + 26 + 32 + 33 + 20 + 29 + + +
+ + + + + +

+ The heights of five plants are measured and recorded below. + Give a 95% confidence interval around the sample mean for the heights of the plants. + (Pretend 5 measurements is large enough for the CLT to apply.) +

+ + + + + + + plant i + 1 + 2 + 3 + 4 + 5 + + + + height \widetilde{h}_i (in) + 15 + 14 + 18 + 21 + 17 + + +
+ + + +
\ No newline at end of file diff --git a/source/sec-Likelihood.ptx b/source/sec-Likelihood.ptx new file mode 100644 index 0000000..f8591fc --- /dev/null +++ b/source/sec-Likelihood.ptx @@ -0,0 +1,60 @@ +
+ Likelihood + +

+ So far, we've been concerned with probability theory. + Starting with a probability distribution and some parameter values, we've tried to answer questions like: What's the probability of seeing certain experimental results? Statistics is concerned with going in the other direction: Upon seeing the experimental results, can we determine the type of underlying probability distribution? Can we determine its parameters? +

+ + + +

+ An estimator is a value of a parameter computed from a sample of data. +

+ + + + +

+ Suppose we find a coin on the street and don't know whether or not it's fair. + We want to know the probability p of the coin coming up heads. + We might, for example, flip the coin n times and count the number k of heads. + Then, we'll estimate p = \frac{k}{n}. + We'll refer to this as a common sense estimator. + (Other distributions and parameter types will have different notions of "common sense".) +

+ + +

+ An estimator is, itself, a random variable: it produces a numerical value based on the results of an experiment. + We'll use notation like \est{p} for a random variable which is an estimator for a parameter p. + (Similarly, \est{\lambda} would denote an estimator for a parameter called \lambda.) +

+ + + +

+ An estimator \est{p} is called unbiased if \E(\est{p}) = p. +

+ + + + + +

+ Suppose we have a coin with parameter p, which we'll flip n times and count the number k of heads. + We use the unbiased estimator \est{p} = \frac{k}{n}. + In this case, notice that k \sim \Bin(n, p), so we know \E(k) = np, although we don't know the value of p. + (We probably do know the value of n; after all, we're flipping the coin!) Now: + + \E(\est{p}) = \E\left(\frac{k}{n}\right) = \frac{1}{n} \cdot \E(k) = \frac{1}{n} \cdot np = p. + + It's worth pausing for a moment to be appropriately impressed with ourselves. + We still don't know the true value of p. + But we managed to show that our common sense method of estimating p gives, on average, the correct value. +

+ + + + +
\ No newline at end of file