### Types I and II errors in significance tests

Type I and type II errors in significance tests

**R computations with normal distributions**

There are various R functions which are useful for computation with normal distributions, such as pnorm( ), qnorm( ), and dnorm( ).

The **p**norm( ) function gives the cumulative distribution function, and the alphabet ‘p’ stands for probability. The qnorm( ) is for **q**uantiles whilst the dnorm( ) function, the **d**ensity.

Let’s use the statistical notation for normal distribution: *X ~ N*(*µ,**sigma** ^{2}*). We shall illustrate the usage of these R functions.

**R function pnorm( )**

For example, let *X ~ N*(8,4), then

(a) the probability *P*(*X *__<__ 2) can be computed via pnorm( ) in several different ways:

> pnorm(2,mean=8,sd=2) #P(X<=2) in N(8,4)

[1] 0.001349898

> pnorm(2,8,2) #P(X<=2) in N(8,4) simplified

[1] 0.001349898

(b) the probability *P*(*X *__<__ 1.96) for x ~ N(8,4) by R language is:

> pnorm(1.96,8,2) #P(X<=1.96) in N(8,4)

[1] 0.001263873

Remember that for *f*(1.96) = 0.975 and *f*(1.645) = 0.950, respectively from the statistics table, the R gives us the same answers:

> pnorm(1.96,0,1) #*P*(X<=1.96) in N(0,1)

[1] 0.9750021

> pnorm(1.645,0,1) #*P*(X<=1.645) in N(0,1)

[1] 0.9500151

> pnorm(1.645) #*P*(X<=1.645) in N(0,1) simplified

[1] 0.9500151

>

And, when *P*(*X *__<__ -1.645), the R result indicates the area on the left hand side of the normal distribution curve:

> pnorm(-1.645) #P(X<=-1.645) in N(0,1) simplified

[1] 0.04998491

>

** ****R function qnorm( )**

In layman’s language, a quantile is where a series of sample data is sub-divided into equal proportions. In statistics, we divide a probability distribution into areas of equal probability. The simplest division that can be envisioned is into two equal halves, i.e., 50%.

The R function: qnorm( ) is used to compute the quantiles for the standard normal distribution using its density function *f*.

For example,

> qnorm(0.95) #95.0% quantile of *N*(0,1)

[1] 1.644854

> qnorm(0.975) #97.5% quantile of *N*(0,1)

[1] 1.959964

>

** ****R function dnorm( )**

The density of a Gaussian formulae for normal distribution can be shown to be close to 0.4 when *x* = 0.

The R function dnorm(0) indeed gives the same result as below:

> dnorm(0) # Density of N(0,1) evaluated at x= 0

[1] 0.3989423

>

**Further remarks**

Like pnorm( ), the functions qnorm( ) and dnorm( ) can also be used for normal distributions with non-zero mean and non-zero standard deviation or variance, simply by supplying the mean and standard deviation as extra arguments.

For example, for the N(8,4) distribution, the results are self-explanatory:

> qnorm(0.975,8,2) # 97.5% quantile of N(8,4)

[1] 11.91993

> dnorm(1,8,2) #Density of N(8,4) at x=1

[1] 0.0004363413

> dnorm(4,8,2) #Density of N(8,4) at x=4

[1] 0.02699548

>

- ANOVA
- Design of Experiments
- Probability
- Significance testing
- Randomization
- Linear regression
- Sampling statistics
- Sampling
- outliers
- Probability distribution
- uncertainty
- Microbiology
- GUM
- Control chart
- Normal distribution
- Measurement error
- Confidence interval
- Degrees of freedom
- Median
- Monte Carlo
- Anderson-Darling
- Risk
- Decision rules
- Conformity testing
- Sampling uncertainty
- F-test
- t-test
- propagation of uncertainty
- p-Value
- Type I and II errors
- ISO 17025
- Decision Rule
- profiency testing
- Confidence limits
- Central limit theorem
- IQR
- Variance
- Law of Averages
- Coverage factor
- How to
- ISO FDIS 17025
- Bias
- Student's t-distribution
- Cross-checks
- Detection limit
- Factorial design
- Chi-square
- interlab comparison
- quartile
- Divisor
- Precision
- Accuracy
- Aerobic plate count
- Compliance
- hypothesis testing
- ISO 17025:2017
- Recovery
- Shewhart
- She
- Method validation
- Excel spreadsheet

## Recent Comments