T Critical Value on a TI-84 Calculator
T Critical Value on a TI-84 Calculator, A t critical value is a cutoff point from the t-distribution used in hypothesis tests and confidence intervals. It depends on two factors:
- Significance level (α)
- Degrees of freedom (df)
For a one-sample t-test, the degrees of freedom are typically: df=n−1
The t critical value is compared with the calculated t-statistic to determine whether there is sufficient evidence to reject the null hypothesis.
What Is a T Critical Value?
The t critical value defines the boundary between the rejection and non-rejection regions of a hypothesis test.
For example, in a two-tailed test with α = 0.05, 5% of the probability is placed in the two tails of the t-distribution: 2α=0.025
If the calculated t-statistic is greater than the positive critical value or less than the negative critical value, the result is statistically significant at the 5% level.
The critical value can also be used when constructing confidence intervals.
How to Find a T Critical Value on a TI-84
The TI-84 uses the invT( function to find a t-distribution quantile.
Step 1: Open the Distribution Menu
On your TI-84 calculator:
- Press 2nd.
- Press VARS to open the DISTR menu.
- Select invT(.
The calculator uses the following syntax:
invT(area, df)
where:
- area = cumulative probability to the left of the desired t-value
- df = degrees of freedom
This distinction is important: for a right-tailed test, you generally use 1 - α, while for a two-tailed test you use α/2 to obtain the negative critical value.
Left-Tailed Test
Suppose you are conducting a left-tailed hypothesis test with:
- α = 0.05
- df = 11
Enter:
invT(.05,11)
The result is approximately: tc=−1.796
Therefore, the rejection region is: t<−1.796
If your calculated t-statistic is less than −1.796, you would reject the null hypothesis at the 5% significance level.
Right-Tailed Test
Suppose:
- α = 0.05
- df = 24
Because invT uses the area to the left, calculate: 1−α=1−0.05=0.95
Enter:
invT(.95,24)
The result is approximately: tc=1.711
The rejection region is therefore: t>1.711
Two-Tailed Test
For a two-tailed test with:
- α = 0.05
- df = 13
Divide the significance level between the two tails: 2α=20.05=0.025
To obtain the negative critical value, enter:
invT(.025,13)
This gives approximately: tc=−2.160
The positive critical value is: tc=2.160
So the two critical values are approximately: −2.160and2.160
The rejection region is: t<−2.160
or t>2.160
Quick Reference Table
| Test | α | df | TI-84 Entry | Approx. Critical Value |
|---|---|---|---|---|
| Left-tailed | 0.05 | 11 | invT(.05,11) | −1.796 |
| Right-tailed | 0.05 | 24 | invT(.95,24) | 1.711 |
| Two-tailed | 0.05 | 13 | invT(.025,13) | ±2.160 |
T Critical Value for a Confidence Interval
T critical values are also frequently used when calculating confidence intervals for a population mean when the population standard deviation is unknown.
For a two-sided confidence interval, the critical value is: tα/2,df
For example, for a 95% confidence interval with 20 degrees of freedom: α=1−0.95=0.05 2α=0.025
On the TI-84, enter:
invT(.975,20)
because the area to the left of the positive critical value is: 1−0.025=0.975
The result is approximately: t∗=2.086
The confidence interval can then be calculated using: xˉ±t∗ns
where:
- xˉ = sample mean
- s = sample standard deviation
- n = sample size
- t∗ = t critical value
T Critical Value vs. Z Critical Value
The t-distribution is generally used when the population standard deviation is unknown and the sample standard deviation is used instead.
The standard normal (Z) distribution is used when the population standard deviation is known or when a Z-based procedure is appropriate.
Unlike the Z critical value, the t critical value changes with the degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.
Common T Critical Values
For a two-tailed test or confidence interval, some commonly used t critical values are:
| Confidence Level | df = 10 | df = 20 | df = 30 |
|---|---|---|---|
| 90% | 1.812 | 1.725 | 1.697 |
| 95% | 2.228 | 2.086 | 2.042 |
| 99% | 3.169 | 2.845 | 2.750 |
These values illustrate why specifying the degrees of freedom is essential when determining a t critical value.
Common Mistakes When Finding T Critical Values
1. Using α instead of 1 − α for a right-tailed test
For a right-tailed test with α = 0.05, use:
invT(.95,df)
not:
invT(.05,df)
The latter produces the negative-side quantile.
2. Forgetting to divide α by 2
For a two-tailed test with α = 0.05: α/2=0.025
For the positive critical value, use:
invT(.975,df)
3. Using the wrong degrees of freedom
For a one-sample t-test: df=n−1
For example, if n=25: df=25−1=24
How to Interpret the Critical Value
Suppose a two-tailed test produces: t=2.45
and the critical values are: −2.160,2.160
Because: ∣2.45∣>2.160
the test statistic falls in the rejection region. At the 5% significance level, you would reject the null hypothesis.
If instead the calculated t-statistic were 1.50, then: ∣1.50∣<2.160
and you would fail to reject the null hypothesis.
Remember that failing to reject the null hypothesis does not prove that the null hypothesis is true. It means that the sample does not provide sufficient evidence against it at the chosen significance level.
Conclusion
Finding a t critical value on a TI-84 calculator is straightforward once you understand the relationship between significance level, tail direction, and degrees of freedom.
The key function is:
invT(area,df)
For a left-tailed test, use the left-tail probability. For a right-tailed test, use 1−α. For a two-tailed test, divide α by 2 and use the appropriate cumulative probability.
Understanding these settings helps you correctly determine rejection regions for hypothesis tests and calculate confidence intervals.
