Z Critical Value on a TI-84 Calculator

Z Critical Value on a TI-84 Calculator, A Z critical value is a value from the standard normal distribution used to determine the rejection region in a hypothesis test or the boundaries of a confidence interval.

On a TI-84 calculator, you can find Z critical values quickly using the invNorm( function.

What Is a Z Critical Value?

A Z critical value is a cutoff point on the standard normal distribution associated with a specified probability or significance level.

For example, at a 95% confidence level, the two-tailed critical values are approximately:

−1.96 and +1.96

In a hypothesis test, the critical value is compared with the calculated test statistic.

For a two-tailed test:

Reject H₀ if |z| > zcritical

For a right-tailed test:

Reject H₀ if z > zcritical

For a left-tailed test:

Reject H₀ if z < zcritical

The critical value depends on whether the test is left-tailed, right-tailed, or two-tailed.

How to Find a Z Critical Value on a TI-84

The TI-84 uses the invNorm( function to find a value associated with a specified cumulative probability.

Step 1: Open invNorm

On your TI-84 calculator:

1. Press 2nd.
2. Press VARS to open the DISTR menu.
3. Select invNorm(.
4. Press ENTER.

The calculator will display:

invNorm(

The general syntax is:

invNorm(area, μ, σ)

For the standard normal distribution:

• area = cumulative probability to the left of the desired Z value
• μ = 0
• σ = 1

Therefore, most Z critical value calculations use:

invNorm(area,0,1)

Example 1: Left-Tailed Test

Suppose you are conducting a left-tailed hypothesis test with:

α = 0.05

Because the rejection region is in the left tail, enter:

invNorm(0.05,0,1)

The TI-84 returns approximately:

zcritical = −1.645

Therefore, the critical region is:

z < −1.645

If your calculated test statistic is less than −1.645, you reject the null hypothesis.

Example 2: Right-Tailed Test

Suppose you are conducting a right-tailed test with:

α = 0.05

The TI-84’s invNorm( function uses the area to the left of the desired value. Therefore, for a right-tailed test, the area to the left is:

1 − α = 1 − 0.05 = 0.95

Enter:

invNorm(0.95,0,1)

The result is approximately:

zcritical = 1.645

Therefore, the rejection region is:

z > 1.645

Example 3: Two-Tailed Test

Suppose you are conducting a two-tailed hypothesis test with:

α = 0.05

The significance level is divided equally between the two tails:

α/2 = 0.05/2 = 0.025

The area to the left of the positive critical value is:

1 − 0.025 = 0.975

Enter:

invNorm(0.975,0,1)

The calculator returns approximately:

zcritical = 1.96

Because the test is two-tailed, the critical values are:

−1.96 and +1.96

The rejection regions are:

z < −1.96

or

z > 1.96

Common Z Critical Values

The following table shows commonly used Z critical values.

For example, for a 99% confidence level:

invNorm(0.995,0,1)

which gives approximately:

2.576

The corresponding two-tailed critical values are:

−2.576 and +2.576

Z Critical Value for Confidence Intervals

Z critical values are also used when constructing confidence intervals.

For a population mean when the population standard deviation is known, a confidence interval can be written as:

x̄ ± zα/2 × σ/√n

where:

• x̄ = sample mean
• zα/2 = Z critical value
• σ = population standard deviation
• n = sample size

For a 95% confidence interval:

zα/2 ≈ 1.96

Therefore:

95% CI = x̄ ± 1.96 × σ/√n

How to Find Z Critical Values for Different Tests

A useful way to remember the TI-84 procedure is to determine the area to the left before using invNorm(.

Z Critical Value vs. Z Test Statistic

It is important not to confuse the Z critical value with the Z test statistic.

The Z test statistic is calculated from your sample data. For example:

z = (x̄ − μ₀)/(σ/√n)

The Z critical value is obtained from the standard normal distribution based on your chosen significance level.

You compare the two values to determine whether the test statistic falls inside or outside the rejection region.

Example: Complete Hypothesis Test

Suppose a company claims that the average battery life of its product is 10 hours.

You want to test:

H₀: μ = 10

H₁: μ ≠ 10

using:

α = 0.05

Because the alternative hypothesis uses ≠, this is a two-tailed test.

Step 1: Find the Critical Value

Divide α between the two tails:

α/2 = 0.025

The area to the left of the positive critical value is:

1 − 0.025 = 0.975

On the TI-84, enter:

invNorm(0.975,0,1)

Result:

zcritical ≈ 1.96

Therefore, the critical values are:

−1.96 and +1.96

Step 2: Compare the Test Statistic

Suppose your calculated test statistic is:

z = 2.31

Because:

2.31 > 1.96

the test statistic falls in the rejection region.

Therefore, you reject H₀ at the 5% significance level.

Important TI-84 Tip

The invNorm( function returns the Z value corresponding to a cumulative area to the left.

This is why the input differs between test types.

For example:

Left-tailed:

invNorm(α,0,1)

Right-tailed:

invNorm(1−α,0,1)

Two-tailed positive critical value:

invNorm(1−α/2,0,1)

This distinction prevents one of the most common mistakes when calculating critical values on a TI-84.

Frequently Asked Questions

What function finds a Z critical value on a TI-84?

Use the invNorm( function.

You can access it through:

2nd → VARS → invNorm(

What is the Z critical value for a 95% confidence level?

For a two-sided 95% confidence interval, the critical values are approximately:

±1.96

What is the Z critical value for α = 0.05?

It depends on the test:

• Left-tailed: −1.645
• Right-tailed: +1.645
• Two-tailed: ±1.960

Why does invNorm(.05,0,1) give a negative value?

Because .05 represents an area of 5% to the left of the Z value. That point lies in the left tail of the standard normal distribution, giving approximately −1.645.

Conclusion

Finding a Z critical value on a TI-84 calculator is straightforward once you understand how the calculator interprets cumulative probability.

The key function is:

invNorm(area,0,1)

Remember that invNorm( uses the area to the left of the desired Z value. For a left-tailed test, use α directly; for a right-tailed test, use 1 − α; and for a two-tailed test, use 1 − α/2 to obtain the positive critical value.

Understanding these steps will help you correctly determine rejection regions for hypothesis tests and calculate confidence intervals using the standard normal distribution.

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