Download Facts and Formulas - Statistical Inference - Study Guide | STAT 431 and more Study notes Statistics in PDF only on Docsity!
Statistics 431: Statistical Inference
Facts and Formulas
Probability foundations
The normal distribution and its samples
- The probability density function of a N (μ, σ 2 ) rv is
f ( x ) =
2 πσ 2
exp
( x − μ)^2 σ 2
- The population mean is μ; the population SD is σ.
- For a sample X 1 ,... , Xn of size n from a normal population, - The sample estimate of μ is the sample mean
X^ ¯ = 1
n
∑^ n
i = 1
Xi.
- The sample estimate of σ 2 is the sample variance
S^2 =
n − 1
∑^ n
i = 1
( Xi − ¯ X )^2 =
n n − 1
[
n
∑^ n
i = 1
X (^) i^2 −
X ¯
]
- The distribution of the normal sample mean: X ¯ ∼ N (μ, σ 2 / n ). The SD of X ¯ , σ/
n , is also called the standard error (SE) of X ¯. We estimate it as S /
n.
- X ¯ and S^2 are independent as rvs. The distribution of the sample variance: ( n − 1 ) S^2 ∼ χ n^2 − 1.
- The sample histogram and the normal quantile plot are two graphical tools to judge whether a sample comes from an approximately normal population. Prefer the quantile plot.
The binomial distribution and its samples
- Let X count the number of “successes” in n independent Bernoulli trials, each with probability p of “success.” Then X has the binomial distribution, X ∼ Bin( n , p ). The probability mass function of a binomial rv is
P ( X = k ) =
n k
pk^ ( 1 − p ) n − k^ , k = 0 ,... , n.
- The population mean is np ; the population SD is
np ( 1 − p ).
- For a binomial rv X based on a Bernoulli sample Z 1 ,... , Zn from a population, the estimate of p is p ˆ = X / n.
- The SD of p ˆ is
p ( 1 − p )/ n ; it is also called the SE of p ˆ. We estimate it as
p ˆ( 1 − ˆ p )/ n.
- For large n , the distribution of p ˆ is approximately N ( p , p ( 1 − p )/ n ).
Chapter 7: Confidence intervals
One sample mean
- A 100γ % = 100 ( 1 − α)% confidence interval (CI) for an unknown population mean μ has the general form
X^ ¯ ± C ∗^ · σ^
∗ √ n
X^ ¯ − C ∗^ · σ^
∗ √ n
, X ¯ + C ∗^ ·
σ ∗ √ n
where C ∗^ is an appropriate upper quantile and σ ∗^ is an appropriate population SD or estimate thereof. The meaning of the confidence statement is that P (μ ∈ Interval) = γ , at least approximately. The important situations are:
- σ known and either population normal or n large: σ ∗^ = σ and C ∗^ = z α/ 2 , the (α/ 2 ) upper quantile of the standard normal. - σ unknown and n large: σ ∗^ = S and C ∗^ = z α/ 2. - σ unknown and population normal: σ ∗^ = S and C ∗^ = t α/ 2 ; n − 1 , the (α/ 2 ) upper quantile of the t distribution with n − 1 degrees of freedom (df).
- The sample size needed to get an interval of width w is (approximately)
n (w) =
2 z α/ 2
σ w
rounded up to the nearest integer. When σ is unknown, use an estimate from previous experience or from the corresponding value of S in a pilot experiment.
One population proportion
- When n is large (say, n p ˆ( 1 − ˆ p ) > 20) and p ˆ is not too near 0 or 1, you can use the classical large-sample CI formula,
p ˆ ± z α/ 2
p ˆ( 1 − ˆ p ) n
and P μ∈ HA (Do not reject H 0 ) = P (Type II error) = β for this μ. The significance level α is the probability of a Type I error at the boundary value μ 0. The power of the test is P μ∈ HA (Reject H 0 )) = 1 − β.
- If we observe the test statistic value T = t , the p -value is the smallest α at which we can reject H 0 using t. If you know the p -value of a test, you know the outcome for every level α:
p -value < α ⇒ reject ; p -value ≥ α ⇒ do not reject.
- Duality: for the usual two-sided tests, a level α test does not reject H 0 : μ = μ 0 exactly when a 100( 1 − α)% CI for μ contains μ 0. (There is a similar relationship between one-sided tests and upper/lower confidence bounds.)
Particular tests: one population mean
- A test of H 0 : μ = μ 0 vs HA : μ 6 = μ 0 rejects when | T | > C ∗, where
T =
X ¯ − μ 0 σ ∗/
n
Here σ ∗^ and C ∗^ are as in the above discussion of two-sided confidence intervals.
- The p -value corresponding to T = t can be found by looking up P (| T | > t ) for the normal distribution (when n is large) or the tn − 1 distribution (when n is small). NOTE: because of the absolute value signs, you must multiply the tabled value by 2.
- The sample size n at which a two-sided level α test has power 1 − β under the alternative μ′ is approximately
n =
σ ∗( z α/ 2 + z β ) μ 0 − μ′
In the one-sided case, put z α in place of z α/ 2. (The resulting n is only valid if it is large, since the formula uses large-sample normality).
- A test of H 0 : μ ≥ μ 0 vs HA : μ < μ 0 would reject when T < C ∗, where C ∗^ is from the lower confidence bound case discussed above. p -values can be found analogously (do not multiply the tabled value by 2).
Particular tests: one population proportion
- To test H 0 : p = p 0 vs HA : p 6 = p 0 , use
T =
p ˆ − p 0 √ p 0 ( 1 − p 0 )/ n
with the critical value determined in the usual manner as a standard normal upper quantile. p -values are also determined from T = t in an analogous manner. Here, n should not be too small; np 0 ( 1 − p 0 ) > 5 should suffice. For smaller n , there is a procedure we have not covered based on the binomial distribution.
- The n for which a two-sided test of p 0 has power 1 − β under the alternative p = p ′^ is approximately
n =
z α/ 2
p 0 ( 1 − p 0 ) + z β
p ′( 1 − p ′) p ′^ − p 0
rounded up to the nearest integer. For a one-sided test, replace z α/ 2 with z α.
Chapter 9: Inferences based on two samples
Inferences about the difference of two population means
- A two-sided hypothesis test has the form H 0 : μ 1 − μ 2 = 10 vs HA : μ 1 − μ 2 6 = 10. Often, 10 = 0. If the sample from population A is independent of the sample from population B, reject when | T | > C ∗, where
T =
X ¯ − ¯ Y − 10
(σ 1 ∗ )^2 / n 1 + (σ 2 ∗ )^2 / n 2
Here σ 1 ∗ , σ 2 ∗ , and C ∗^ are like the values in the one-sample procedures. However, if n 1 or n 2 is small and σ is unknown, you need to assume normal population distributions and treat T as having a t distribution with ν df. Here
ν =
S^21 n 1 +^
S 22 n 2
( S 12 / n 1 )^2 n 1 − 1 +^
( S 22 / n 2 )^2 n 2 − 1
rounded to the nearest integer. Note that min{ n 1 − 1 , n 2 − 1 } ≤ ν ≤ n 1 + n 2 + 1.
- A 100( 1 − α)% CI for μ 1 − μ 2 takes the form X ¯ − ¯ Y ± C ∗^ · SE, where SE is the denominator of the test statistic T.
- p -values can be found in the usual way from the value T = t and the corresponding table.
- Formulas for β can be derived from the structure of the test. Samples size calculations are complicated, except in special cases. We do not give general formulas here, or for two proportions below.
- If the additional assumption σ 1 = σ 2 is tenable, use
T =
X ¯ − ¯ Y − 10
S^2 pooled( 1 / n 1 + 1 / n 2 )
