How to Calculate the P-Value of a Z-Score in R

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Often in statistics we’re interested in determining the p-value associated with a certain z-score that results from a hypothesis test. If this p-value is below some significance level, we can reject the null hypothesis of our hypothesis test.

To find the p-value associated with a z-score in R, we can use the pnorm() function, which uses the following syntax:

pnorm(q, mean = 0, sd = 1, lower.tail = TRUE)

where:

• q: The z-score
• mean: The mean of the normal distribution. Default is 0.
• sd: The standard deviation of the normal distribution. Default is 1.
• lower.tail: If TRUE, the probability to the left of q in the normal distribution is returned. If FALSE, the probability to the right is returned. Default is TRUE.

The following examples illustrate how to find the p-value associated with a z-score for a left-tailed test, right-tailed test, and a two-tailed test.

Left-tailed test

Suppose we want to find the p-value associated with a z-score of -0.77 in a left-tailed hypothesis test.

#find p-value
pnorm(q=-0.77, lower.tail=TRUE)

[1] 0.2206499

The p-value is 0.2206. If we use a significance level of α = 0.05, we would fail to reject the null hypothesis of our hypothesis test because this p-value is not less than 0.05.

Right-tailed test

Suppose we want to find the p-value associated with a z-score of 1.87 in a right-tailed hypothesis test.

#find p-value
pnorm(q=1.87, lower.tail=FALSE)

[1] 0.03074191

The p-value is 0.0307. If we use a significance level of α = 0.05, we would reject the null hypothesis of our hypothesis test because this p-value is less than 0.05.

Two-tailed test

Suppose we want to find the p-value associated with a z-score of 1.24 in a two-tailed hypothesis test.

#find p-value for two-tailed test
2*pnorm(q=1.24, lower.tail=FALSE)

[1] 0.2149754

To find this two-tailed p-value we simply multiplied the one-tailed p-value by two.

The p-value is 0.2149. If we use a significance level of α = 0.05, we would fail to reject the null hypothesis of our hypothesis test because this p-value is not less than 0.05.

Related: You can also use this online Z Score to P Value Calculator to find p-values.