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


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 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.

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