How to Find Z Critical Values in R

When we conduct a hypothesis test, we obtain test statistics as an outcome.  Now in order to find out whether the outcome of the hypothesis test is statistically significant, the Z critical value is compared with the test statistic.  If the absolute value of the test statistic comes out to be greater than the Z critical value then the outcome of the hypothesis test is considered statistically significant. 

Determining the Z critical values in R:

R provides us the qnorm() function using which we can determine the Z critical values in R. The function has the following syntax:

qnorm(p, mean = 0, sd = 0, lower.tail = TRUE)

Here,

• p: It represents the significant level to be used
• mean: It represents the mean of the normal distribution
• sd: It represents the standard deviation of the normal distribution
• lower.tail = TRUE: Then the probability to the left of p in the normal distribution is returned. 
• lower.tail = TRUE: Then the probability to the right is returned. 
• Note that by default lower.tail is TRUE.

Now let's discuss how we can determine the Z critical value for the left-tailed test, a right-tailed test, and a two-tailed test.

Left-tailed test:

A left-tailed test is used when the alternative hypothesis states that the true value of the parameter specified in the null hypothesis is less than the null hypothesis claims.

Suppose that we want to determine the Z critical value for a left-tailed test having a significance level equal to 0.01 by passing the p-value to 0.01 and the lower.tail value to TRUE since for the left-tailed test within the qnorm() function in the R programming language.

Example:

# Determine the Z critical value
qnorm(p=.01, lower.tail=TRUE)

Output:

Interpretation of output:

The Z critical values come out to be equal to -2.326. Therefore, if the test statistics comes out to be less than this value then the outcome of the hypothesis test would be considered statistically significant.

Right-tailed test:

A right-tailed test is used when the alternative hypothesis states that the true value of the parameter specified in the null hypothesis is greater than the null hypothesis claims.

Suppose that we want to determine the Z critical value for a right-tailed test having a significance level equal to 0.01 bypassing the p-value to 0.01  and the lower.tail to false due to the right-tailed test of the qnorm function in the R.

# Determine the Z critical value
qnorm(p=.01, lower.tail=FALSE)

Output:

Interpretation of output:

The Z critical values come out to be equal to 2.326. Therefore, if the test statistics comes out to be greater than this value then the outcome of the hypothesis test would be considered statistically significant.

Two-tailed test:

A two-tailed test, is a method in which the critical area of a distribution is two-sided and tests whether a sample is greater than or less than a certain range of values.

Suppose that we want to determine the Z critical value for a two-tailed test having a significance level equal to 0.01, we are passing the value to the p argument of the qnorm function to 0.01/2 and the lower.tail value of false since it is the two-tailed test in R.

# Determine Z critical value
qnorm(p=.01/2, lower.tail=FALSE)

Output:

Interpretation of output:

When we conduct the two-tailed test, we get two critical values. Here, the two Z critical values come out to be equal to 2.575 and -2.575. Therefore, if the test statistics comes out to be lesser than -2.575 or greater than 2.575 then the outcome of the hypothesis test would be considered statistically significant.