# How to find critical values and p-values from the normal distribution

## Description

Some statistical techniques require computing critical values or $p$-values from the normal distribution. For example, we need to do this when constructing a confidence interval or conducting a hypothesis test. How do we compute such values?

## Solution, in Julia

View this solution alone.

If we choose a value $0 \le \alpha \le 1$ as our Type 1 error rate, then we can find the critical value from the normal distribution using the quantile() function in Julia’s Distributions package.

If you don’t have that package installed, first run using Pkg and then Pkg.add( "Distributions" ) from within Julia.

The code below shows how to do this for left-tailed, right-tailed, and two-tailed hypothesis tests.

1
2
3
4
using Distributions
alpha = 0.05                            # Replace with your alpha value
standard_normal = Normal( 0, 1 )
quantile( standard_normal, alpha )      # Critical value for a left-tailed test

1
-1.6448536269514724

1
quantile( standard_normal, 1 - alpha )  # Critical value for a right-tailed test

1
1.6448536269514717

1
quantile( standard_normal, alpha / 2 )  # Critical value for a two-tailed test

1
-1.9599639845400592


We can also compute $p$-values from the normal distribution to compare to a test statistic. As an example, we’ll use a test statistic of 2.67, but you can substitute your test statistic’s value instead.

We can find the $p$-value for this test statistic using the cdf() function in Julia’s Distributions package. Again, we show code for left-tailed, right-tailed, and two-tailed tests.

1
2
test_statistic = 2.67                               # Replace with your test statistic
cdf( standard_normal, test_statistic )              # p-value for a left-tailed test

1
0.9962074376523146

1
1 - cdf( standard_normal, test_statistic )          # p-value for a right-tailed test

1
0.0037925623476854353

1
2 * ( 1 - cdf( standard_normal, test_statistic ) )  # p-value for a two-tailed test

1
0.007585124695370871


See a problem? Tell us or edit the source.

## Solution, in R

View this solution alone.

If we choose a value $0 \le \alpha \le 1$ as our Type 1 error rate, then we can find the critical value from the normal distribution using R’s qnorm() function. The code below shows how to do this for left-tailed, right-tailed, and two-tailed hypothesis tests.

1
2
3
4
alpha <- 0.05                           # Replace with your alpha value
qnorm(p = alpha, lower.tail = TRUE)     # Critical value for a left-tailed test
qnorm(p = alpha, lower.tail = FALSE)    # Critical value for a right-tailed test
qnorm(p = alpha/2, lower.tail = FALSE)  # Critical value for a two-tailed test

1
2
3
4
5
6
7
8
9
[1] -1.644854

[1] 1.644854

[1] 1.959964


We can also compute $p$-values from the normal distribution to compare to a test statistic. As an example, we’ll use a test statistic of 2.67, but you can substitute your test statistic’s value instead.

We can find the $p$-value for this test statistic using R’s pnorm() function. Again, we show code for left-tailed, right-tailed, and two-tailed tests.

1
2
3
4
test_statistic <- 2.67                       # Replace with your test statistic
pnorm(test_statistic, lower.tail = TRUE)     # p-value for a left-tailed test
pnorm(test_statistic, lower.tail = FALSE)    # p-value for a right-tailed test
2*pnorm(test_statistic, lower.tail = FALSE)  # p-value for a two-tailed test

1
2
3
4
5
6
7
8
9
[1] 0.9962074

[1] 0.003792562

[1] 0.007585125