Term | Description |
---|---|
rate of occurrence for sample i |
Term | Description |
---|---|
mean number of occurrences in sample i |
The normal approximation test is based on the following Z-statistic, which is approximately distributed as a standard normal distribution under the null hypothesis:
Minitab uses the following p-value equations for the respective alternative hypotheses:
Term | Description |
---|---|
observed value of rate for sample X | |
observed value of rate for sample Y | |
ζ | true value of the difference between the population rates of two samples |
ζ_{0} | hypothesized value of the difference between the population rates of two samples |
m | sample size of sample X |
n | sample size of sample Y |
t_{x} | length of sample X |
t_{y} | length of sample Y |
When the hypothesized difference equals 0, Minitab uses an exact procedure to test the following null hypothesis:
H_{0}: ζ = λ_{x} – λ_{y} = 0, or H_{0}: λ_{x} = λ_{y}
The exact procedure is based on the following fact, assuming the null hypothesis is true:
S | W ~ Binomial(w, p)
where:
W = S + U
H_{1}: ζ > 0: p-value = P(S ≥ s | w = s + u, p = p_{0})
H_{1}: ζ < 0: p-value = P(S ≤ s | w = s + u, p = p_{0})
then the p-value = 2 × min {P(S ≤ s | w = s + u, p = p_{0}), P(S ≥ s | w = s + u, p = p_{0})}
where:
Term | Description |
---|---|
observed value of the rate for sample X | |
observed value of the rate for sample Y | |
λ_{x} | true value of the rate for population X |
λ_{y} | true value of rate for population Y |
ζ | true value of the difference between the population rates of two samples |
t_{x} | length of sample X |
t_{y} | length of sample Y |
m | sample size of sample X |
n | sample size of sample Y |
When you test a zero difference with the following null hypothesis, you have the option to use a pooled rate for both samples:
The pooled-rate procedure is based on the following Z-statistic, which is approximately distributed as a standard normal distribution under the following null hypothesis:
where:
Minitab uses the following p-value equations for the respective alternative hypotheses:
Term | Description |
---|---|
observed value of the rate for sample X | |
observed value of the rate for sample Y | |
λ_{x} | true value of the rate for population X |
λ_{y} | true value of the rate for population Y |
ζ | true value of the difference between the population rates of two samples |
m | sample size of sample X |
n | sample size of sample Y |
t_{x} | length of sample X |
t_{y} | length of sample Y |
The normal approximation test is based on the following Z-statistic, which is approximately distributed as a standard normal distribution under the null hypothesis.
Minitab uses the following p-value equations for the respective alternative hypotheses:
Term | Description |
---|---|
observed value of the mean number of occurrences in sample X | |
observed value of the mean number of occurrences in sample Y | |
δ | true value of the difference between the population means of two sample |
δ _{0} | hypothesized value of the difference between the population means of two samples |
m | sample size of sample X |
n | sample size of sample Y |
The exact procedure is based on the following fact, assuming the null hypothesis is true:
S | W ~ Binomial(w, p)
where:
W = S + U
Minitab uses the following p-value equations for the respective alternative hypotheses:
H_{1}: δ > 0: p-value = P(S ≥ s | w = s + u, δ = 0)
H_{1}: δ < 0: p-value = P(S ≤ s | w = s + u, δ = 0)
if P(S ≤ s|w = s + u, δ = 0) ≤ 0.5
or P(S ≥ s|w = s + u, δ = 0) ≤ 0.5
then:
A two-tailed test is not an equal-tailed test unless m = n.
Term | Description |
---|---|
μ_{x} | true value of the mean number of occurrences in population X |
μ_{y} | true value of the mean number of occurrences in population Y |
δ | true value of the difference between the population means of two samples |
m | sample size of sample X |
n | sample size of sample Y |
The pooled-mean procedure is based on the following Z-value, which is approximately distributed as a standard normal distribution under the following null hypothesis:
where:
Minitab uses the following p-value equations for the respective alternative hypotheses:
Term | Description |
---|---|
observed value of the mean number of occurrences in sample X | |
observed value of the mean number of occurrences in sample Y | |
µ_{x} | true value of the mean number of occurrences in population X |
µ_{y} | true value of the mean number of occurrences in population Y |
δ | true value of the difference between the population means of two samples |
m | sample size of sample X |
n | sample size of sample Y |
A 100(1 – α)% confidence interval for the difference between two population Poisson rates is given by:
Term | Description |
---|---|
observed value of rate for sample X | |
observed value of rate for sample Y | |
ζ | true value of the difference between the population rates of two samples |
z_{x} | upper x percentile point of the standard normal distribution, where 0 < x < 1 |
m | sample size of sample X |
n | sample size of sample Y |
t_{x} | length of sample X |
t_{y} | length of sample Y |
When you specify a "greater than" test, a 100(1 – α)% lower confidence bound for the difference between two population Poisson rates is given by:
When you specify a "less than" test, a 100(1 – α)% upper confidence bound for the difference between two population Poisson rates is given by:
Term | Description |
---|---|
observed value of rate for sample X | |
observed value of rate for sample Y | |
ζ | true value of the difference between the population rates of two samples |
z_{x} | the upper x percentile point on the standard normal distribution, where 0 < x < 1 |
m | sample size of sample X |
n | Sample size of sample Y |
t_{x} | length of sample X |
t_{y} | length of sample Y |
A 100(1 – α)% confidence interval for the difference between two population Poisson means is given by:
Term | Description |
---|---|
observed value of the mean number of occurrences in sample X | |
observed value of the mean number of occurrences in sample Y | |
δ | true value of the difference between the population means of two samples |
z_{x} | upper x percentile point on the standard normal distribution, where 0 < x < 1 |
m | sample size of sample X |
n | sample size of sample Y |
When you specify a "greater than" test, a 100(1 – α)% lower confidence bound for the difference between two population Poisson means is given by:
When you specify a "less than" test, a 100(1 – α)% upper confidence bound for the difference between two population Poisson means is given by:
Term | Description |
---|---|
observed value of the mean number of occurrences in sample X | |
observed value of the mean number of occurrences in sample Y | |
δ | true value of the difference between the population means of two samples |
z_{x} | upper x percentile point on the standard normal distribution, where 0 < x < 1 |
m | sample size of sample X |
n | sample size of sample Y |