






Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Definitions and formulas are laws of probability, theoretical mean variance for discrete distributions, correlation and regression, analysis of variance and median test for two independent samples. From university of oxford.
Typology: Cheat Sheet
1 / 12
This page cannot be seen from the preview
Don't miss anything!
6 Standard errors
Single sample of size n
SE(¯x) = √σn or, if σ unknown, √sn
SE(ˆp) =
n with^ q^ = 1^ −^ p^ ,^ or, if^ p^ unknown,
pˆ(1−ˆp) n
Sampling without replacement
When n individuals are sampled from a population of N without replacement, the standard error is reduced. The standard error for no replacement SENR is related to the standard error with replacement SEWR by the formula
n − 1 N − 1
σ √ n
n − 1 N − 1
where σ is the known standard deviation of the whole population.
Two independent samples of sizes, n 1 and n 2
SE(¯x 1 − x¯ 2 ) =
σ 12
σ^22 n 2 or, if^ σ^1 and^ σ^2 unknown and different,
s^21
s^22 n 2.
1
1 n 2 with^ s
(^2) =(n^1 −1)s (^21) +(n 2 −1)s (^22) n 1 +n 2 − 2
SE(ˆp 1 − pˆ 2 ) =
p 1 q 1
p 2 q 2 n 2 ,^ or,
if p 1 and p 2 unknown and unequal,
pˆ 1 ˆq 1
pˆ 2 ˆq 2 n 2
For common but unknown p , SE(ˆp 1 − pˆ 2 ) =
1 +n 2 and ˆq = 1 − pˆ.
7 95% confidence limits for population parameters
Mean: when σ known use ¯x ± 1. 96 √σn
where t is the tabulated two-sided 5% level value with degrees of freedom, d.f. = n − 1
Proportion: pˆ ± 1. 96
pˆq/nˆ
8 z-tests
Single sample test for population mean μ (known σ): z = ¯x−μ σ/ √ n
n
σ^21 n 1 +^
σ^22 n 2
Two sample test for difference between two proportions : z = √ pˆ^1 −^ pˆ^2 p ˆqˆ( (^) n^11 + (^) n^12 )
n 1 pˆ 1 +n 2 pˆ 2 n 1 +n 2 and ˆq^ = 1^ −^ pˆ
9 t-tests
Population variance σ^2 unknown and estimated by s^2
Single sample test for population mean μ t = x¯−μ s/
n with d.f.= n − 1
Paired samples : test for zero mean difference, using n pairs (x, y), d = x − y
t = d¯ sd/
n
Independent samples test for difference between population means μx and μy using nx x’s and ny y’s. Provided that s^2 x and s^2 y are similar values, use the pooled variance estimate,
s^2 =
(nx−1)s^2 x+(ny−1)s^2 y nx+ny− 2 ,^ and^ t^ =^
¯x−¯y s
1 nx +^
1 ny
with d.f = nx + ny − 2
10 The χ^2 -test
(Note that the two tests in this section are nonparametric tests. There are χ^2 tests of vari- ances, not included here, that are parametric.)
χ^2 Goodness-of-fit tests using k groups have d.f.=(k − 1) − p where p is the number of independent parameters estimated and used to obtain the (fitted) expected values.
χ^2 Contingency table tests on two-way tables with r rows and c columns have d.f.= (r − 1)(c − 1)
(O−E)^2 E where^ O^ is an observed frequency and^ E^ is the corresponding expected frequency
13 Median test for two independent samples
For two independent samples, sizes n 1 and n 2 , the median of the whole sample of n = n 1 + n 2 observations is found. The number in each sample above this median is counted and expressed as a proportion of that sample size. The two proportions are compared using the Z-test as in §8.
14 Rank sum test or Mann-Whitney test
For two independent samples, sizes n 1 and n 2 , ranked without regard to sample, call the sum of the ranks in the smaller sample R. If n 1 ≤ n 2 ≤ 10 refer to Table 5, otherwise use a Z
test with z = (R − μ)/σ where μ = 12 n 1 (n 1 + n 2 + 1) and σ =
√ n 1 n 2 (n 1 +n 2 +1) 12 , assuming n 1 ≤ n 2. In case of ties, ranks are averaged.
15 Sign test for matched pairs
The number of positive differences from the n pairs is counted. This number is binomially distributed with p = 12 , assuming a population zero median difference. So apply the Z test for a binomial proportion with p = 12.
16 Wilcoxon test for matched pairs
Ignoring zero differences, the differences between the values in each pair are ranked without regard to sign and the sums of the positive ranks, R+ and of the negative ranks, R−, are calculated. (Check R+ + R− = 12 n(n + 1), where n is the number of nonzero differences). The smaller of R+ and R− is called T and may be compared with the critical values in Table 6 for a two-tailed test. (For one-tailed tests, use R− and R+ with the same table, remembering to halve P .) In case of ties, ranks are averaged.
17 Kolmogorov-Smirnov test
Two samples of sizes n 1 and n 2 are each ordered along a scale. At each point on the scale the empirical cumulative distribution function is calculated for each sample and the difference between the pairs are recorded as Di. The largest absolute value of the Di is called Dmax and this value is compared with the 5% one-tailed value
Dcrit = 1. 36
n 1 + n 2 n 1 n 2
Single sample version, compares sample with theoretical distribution,
Dcrit = 1. 36
n
Should only be used with no ties, but it commonly is used otherwise. With ties, the value of Dmax tends to be too small, so that the p-value is an overestimate.
18 Kruskal-Wallis test for several independent samples
(Analysis of variance for a single factor). For k samples of sizes n 1 , n 2 , ..nk, comprising a total of n observations, all values are ranked without regard to sample, from 1 to n. The rank sums for the samples are calculated as R 1 , R 2 , .., Rk. (Check ΣRi = 12 n(n + 1). The test statistic is
[ 12 n(n+1)Σ
R i^2 ni
] −3(n + 1),
which is compared to χ^2 table with d.f. = k − 1
19 Spearman’s Rank Correlation Coefficient
If x and y are ranked variables the Spearman Rank Correlation Coefficient is just the sample product moment correlation coefficient between the pairs of ranks, rs, which may also be computed by
rs = 1− 6Σd^2 n(n^2 −1)
where d is the difference x − y, and n is the number of pairs (x, y).
Test rs using t = rs
√n−^2 1 −r^2 s
with d.f.= n − 2
2 !
Variance ratio F = s^21 /s^22 with ν 1 and ν 2 degrees of freedom respectively.
23 TABLE 5 : Critical values of R for the Mann-Whitney rank-sum test The pairs of values below are approximate critical values of R for two-tailed tests at levels P = 0.10 (upper pair) and P = 0.05 (lower pair). (Use relevant P = 0.10 entry for one-tailed