P-Value From Z Calculator

How to Use the P-Value From Z Calculator

This calculator helps you find a p-value when you already have the z-statistic from a hypothesis test. To use the calculator:

1. Enter the z statistic, z. The calculator also accepts positive, negative, or zero z statistics. For example, you can enter 2.15, -1.84, or 0.
2. Click Calculate.

The calculator instantly returns the left-tailed, right-tailed, and two-tailed p-value. You can then select the p-value that matches your alternative hypothesis.

Want to learn how to get a p-value from a z-statistic using tables? The calculator also provides a clear, step-by-step solution explaining how each of the p-values can be calculated using a z-score table.

What Is a P-Value From a Z Score?

A p-value from a z score is the probability of obtaining a z statistic at least as extreme as the observed value, assuming the null hypothesis is true.

The z statistic tells us where the observed test result falls on the standard normal distribution. Once you know z, you can use the area under the standard normal curve to find the p-value.

Unlike a t statistic, a z statistic does not require degrees of freedom. Instead, we use a standard normal distribution, which has a mean of 0 and a standard deviation of 1.

The only additional information you need is the direction of the alternative hypothesis so that you know which p-value to use.

How to Convert a Z Score to a P Value

To convert a z score to a p-value, we use the cumulative standard normal distribution. Let Φ(z) = P(Z ≤ z). In this case, Φ(z) is the area under the standard normal curve to the left of the observed z statistic.

Once you know the area to the left of the standard normal table, you can calculate the p-value as follows:

• Left-Tailed P-Value. This is simply the area to the left of the observed z statistic. In other words, left-tailed p-value = Φ(z) or P(Z ≤ z).
• Right-Tailed P-Value. This is the area to the right of a standard normal table. Therefore, right-tailed p-value = 1 − Φ(z) or P(Z ≥ z)
• Two-Tailed P-Value. For a two-tailed hypothesis test, we consider equally extreme values on both sides of the standard normal distribution. Therefore, two-tailed p-value = 2 × P(Z ≥ |z|). Alternatively, to find the two-tailed p-value from a z-statistic, find the minimum of the left-tailed and right-tailed p-values and multiply it by 2. That is. two-tailed p-value = 2 × min[Φ(z), 1 − Φ(z)]

Example 1: Find a Right-Tailed P-Value

A school district tests whether a new tutoring program increases the mean mathematics score of high school students. The test produces a z-statistic of 2.24. Find the p-value for the test.

Solution

Since we want to test whether the new tutoring program increases the mean mathematics score, this is a right-tailed test. We also know that the z-statistic is z = 2.24.

To find the right-tailed p-value for the test, we use a Z table and find P(Z > 2.24).

Recall that a Z table gives us the area to the left of a z-score. Thus, when we look up 2.2 in the row and 0.04 in the column, the two values meet at 0.9875. This corresponds to: P(Z < 2.24) = 0.9875

However, we need to find P(Z > 2.24).

Using the complement rule, P(Z > 2.24) = 1 – P(Z < 2.24)

= 1 – 0.9875

= 0.0125

Therefore, the right-tailed p-value for the test is 0.0125. The calculator gives a more precise right-tailed p-value of 0.012545.

Now, assuming a 5% significance level, we would reject the null hypothesis, H₀, because the p-value, 0.0125, is less than the significance level, 0.05.

Therefore, at the 5% significance level, there is sufficient evidence to support the claim that the tutoring program increases the mean mathematics score.

Example 2: Find a Left-Tailed P-Value

A battery manufacturer claims that its batteries have a mean operating life of 12 hours. A quality-control analyst wants to determine whether the true mean battery life has fallen below 12 hours. The test produces a z-statistic of -1.73. Find the p-value for the test.

Solution

Since we want to test whether the mean battery life has fallen below 12 hours, this is a left-tailed test. We also know that the z-statistic is z = -1.73.

To find the left-tailed p-value for the test, we use a Z table and find P(Z < -1.73).

Recall that a Z table gives us the area to the left of a z-score. Thus, when we look up -1.7 in the row and 0.03 in the column, the two values meet at 0.0418. This corresponds directly to P(Z < -1.73) = 0.0418

Since this is a left-tailed test, the area given by the Z table is already the p-value.

Therefore, the left-tailed p-value for the test is 0.0418. The calculator gives a more precise left-tailed p-value of 0.041815.

Now, assuming a 5% significance level, we would reject the null hypothesis, H₀, because the p-value, 0.0418, is less than the significance level, 0.05.

Therefore, at the 5% significance level, there is sufficient evidence to support the claim that the mean battery life has fallen below 12 hours.

Example 3: Find a Two-Tailed P-Value

A community college states that students spend an average of 8 hours per week using its online learning system. A researcher wants to determine whether the actual mean weekly usage differs from 8 hours. The test produces a z-statistic of -2.08. Find the p-value for the test.

Solution

Since we want to test whether the mean weekly usage differs from 8 hours, this is a two-tailed test. We also know that the z-statistic is z = -2.08.

To find the two-tailed p-value for the test, we first use a Z table to find the area in the smaller tail.

Recall that a Z table gives us the area to the left of a z-score. Thus, when we look up -2.0 in the row and 0.08 in the column, the two values meet at 0.0188. This corresponds to, P(Z < -2.08) = 0.0188

Since this is a two-tailed test, we need to account for equally extreme values in both tails of the standard normal distribution.

Therefore, the two-tailed p-value = 2 × P(Z < -2.08)

= 2 × 0.0188

= 0.0376

Therefore, the two-tailed p-value for the test is 0.0376. The calculator gives a more precise two-tailed p-value of 0.037526.

Now, assuming a 5% significance level, we would reject the null hypothesis, H₀, because the p-value, 0.0376, is less than the significance level, 0.05.

Therefore, at the 5% significance level, there is sufficient evidence to conclude that the mean weekly usage differs from 8 hours.

P-Value From Z vs. Critical Z Value

The p-value method and critical-value method are two ways of making decisions about the null hypothesis. Both methods should arrive at the same decision and conclusion provided you apply them correctly.

With the p-value approach:

1. Calculate the z statistic.
2. Convert the z statistic to a p-value.
3. Compare p with the significance level, α.

The common decision rule is:

• If p < α, reject H₀.
• If p ≥ α, fail to reject H₀.

However, with the critical-value approach:

1. Calculate the z-statistic
2. Choose the significance level.
3. Find the critical z value.
4. Compare the calculated z statistic with the critical boundary.

In this case, the decision rule is as follows:

• If the absolute z-statistic is greater than the absolute z-critical value, we reject H₀.
• If the absolute z-statistic is less than or equal to the absolute z-critical value, we fail to reject H₀.

When Should You Use a Z Score to P Value Calculator?

Use a z score to p value calculator when your statistical procedure produces a z test statistic, and you need the corresponding p-value.

Common examples include:

• one-sample z tests for a population mean when σ is known;
• one-proportion z test
• two-proportion z test;
• large-sample hypothesis tests that use the standard normal distribution;
• normal-approximation test statistics.

If your test produces another statistic, such as t, F, chi-square, or Pearson r, use the general P-Value Calculator instead.

Common Mistakes When Converting Z to a P-Value

Here are some common errors to avoid:

• Using the wrong tail. Determine the test direction from H₁ before choosing the final p-value.
• Ignoring a negative sign. z = -2.10 and z = 2.10 have different left- and right-tailed p-values.
• Doubling the wrong probability. For a two-tailed test, double the smaller tail probability.
• Using a raw score as the z statistic. A Z table and this calculator require a standardized z statistic.
• Treating the Z-table value as a right-tail probability. Our standard normal table reports cumulative probability to the left.
• Assuming α is always 0.05. Use the significance level specified for the hypothesis test.
• Reporting p = 0 for an extreme z statistic. The true probability may be extremely small without being exactly zero.

Frequently Asked Questions