Summary of Statistical Inference for Values of Population Parameters
Parameter
/ Population mean, m / Population proportion, pPoint Estimator
/ Sample mean, / Sample proportion,Inferential Statistic for Hypothesis Testing / The value m0 is the value specified in the null hypothesis.
, when the population is normal, or when n is large. Under the null hypothesis, this statistic has a t distribution with n – 1 degrees of freedom. / The value p0 is the value specified in the null hypothesis.
when n is large enough, and when np ³ 5 and
n(p – 1) ³ 5. Under the null hypothesis, this statistic has an approximate standard normal distribution.
Approximate
(1- a)100% Confidence Interval for the Parameter Value / , when the population is normal, or n is large. /
Parameter
/ Population variance, / Ratio of two independent population variances,Point Estimator
/ Sample variance, S2 / Ratio of the sample variances,Inferential Statistic for Hypothesis Testing / The value is the value specified in the null hypothesis.
. Under the null hypothesis, this statistic has a chi-square distribution with d.f. = n - 1. Here it is assumed that the sample is taken from a Normal(m, ) distribution. / , assuming that both samples come from normal distributions. Here the null hypothesis is equality of the population variances.
Approximate
(1- a)100% Confidence Interval for the Parameter Value / / . Here again, it is assumed that both samples are from normal populations.
Parameter
/ Difference between two independent population means, m1 - m2 / Difference between two independent population proportions, p1 – p2Point Estimator
/ Difference between the sample means, / Difference between the sample proportions,Inferential Statistic for Hypothesis Testing / The value d0 in these formulas is the difference specified in the null hypothesis (usually, d0 = 0).
, when s1 and s2 are unknown, but are equal to each other and the populations are normal. Under the null hypothesis, this statistic has a t distribution with n1 + n2 – 2 degrees of freedom.
Here .
If the variances are not assumed to be equal, then the test statistics has the form:
, which under H0 has an approximate t distribution with approximate degrees of freedom:
/ The value d0 in this formula is the difference specified in the null hypothesis (here, d0 ¹ 0).
when n1 and n2 are large enough, and when n1p1 ³ 5,
n2p2 ³ 5, n1(1-p1) ³ 5, and n2(1-p2) ³ 5. Under the null hypothesis, this statistic has an approximate standard normal distribution.
If the number appearing in the null hypothesis is 0, the following test statistic is used:
where . Under the null hypothesis, this statistic has an approximate standard normal distribution.
Approximate
(1 - a)100% Confidence Interval for the Parameter Value / /