What a Z-Test Is and When to Use It on the TI-84
A Z-test evaluates whether a sample mean is significantly different from a known population mean when the population standard deviation is known and the sample size is large or the population is normal. On the TI-84, the “1-PropZTest” or “Z-Test” function standardizes the difference between your sample mean and the hypothesized mean using the population standard deviation, then returns a test statistic and p-value. Use a Z-test when the population variance is known or can be reasonably assumed, your data are continuous and approximately normal (or n ≥ 30 by the Central Limit Theorem), and your sample is random and independent.
Before you begin, verify that your problem statement provides the population mean (μ), the population standard deviation (σ), the sample mean (x̄), and the sample size (n). If you do not know the population standard deviation and are using the sample standard deviation (s), consider using a T-test instead. The TI-84’s Z-test function produces the z-statistic, the p-value for a left-tailed, right-tailed, or two-tailed alternative, and the sample mean compared to the hypothesized mean.
How to Access the Z-Test Function on TI-84
Open the Stat Tests Menu
Press STAT, then use the right arrow to move to the TESTS menu. On many TI-84 models, you can scroll directly to Z-Test…. If you are performing a one-proportion test, you would choose 1-PropZTest instead; for a one-sample t-test with unknown population standard deviation, choose T-Test….
Input Data Either Directly or from a List
On the Z-Test screen you can enter summary statistics on the main line or input stored list data. For summary stats, select Stats, then enter:
- μ0: the hypothesized population mean
- x̄: the sample mean
- σ: the population standard deviation
- n: the sample size
Alternatively, if your sample data are stored in a list (e.g., L1), select Data and set List to that list. For the Z-test, the calculator will use the stored list values and the given μ0 and σ to compute the test statistic.
Step-by-Step Procedure to Conduct a Z-Test
- State hypotheses. Define H₀: μ = μ0 versus Hₐ: μ ≠ μ0 (two-tailed), μ μ0 (right-tailed).
- Check conditions. Ensure either the population is normal or n ≥ 30 for the Central Limit Theorem, and that the sample is random and independent. Confirm σ is known.
- Gather calculator inputs. Determine μ0, x̄, σ, n, and the tail type.
- Open Z-Test. Press STAT → TESTS → Z-Test….
- Enter summary stats or list. Choose Stats or Data and input μ0, x̄, σ, n, and select the appropriate Tail.
- Calculate. Highlight Calculate and press ENTER.
- Interpret output. The calculator will display z (test statistic), p-value, and sample mean x̄. Compare the p-value to your significance level (commonly α = 0.05) to decide whether to reject H₀.
Interpreting Z-Test Output from the TI-84
The test statistic z measures how many standard errors the sample mean is from the hypothesized mean. A larger absolute z indicates stronger evidence against the null. The p-value reflects the probability of observing a sample mean as extreme as, or more extreme than, the one observed if H₀ were true. For a two-tailed test, double the one-tail p-value returned if needed. If p ≤ α, reject H₀; otherwise, do not reject H₀. In addition to the decision, report the context, the z-statistic, the p-value, and a conclusion in plain language.
Example: Testing a Hypothesized Mean with Known Sigma
Suppose a manufacturer claims a machine fills bottles to a mean of 500 mL with a known standard deviation of 8 mL. A random sample of 36 bottles yields x̄ = 506 mL. To test H₀: μ = 500 mL versus Hₐ: μ ≠ 500 mL at α = 0.05:
- Conditions: n = 36 ≥ 30, random sample, σ known.
- Inputs: μ0 = 500, x̄ = 506, σ = 8, n = 36, tail = two-tailed.
- Calculator output might show z ≈ 4.50 and p ≈ 0.000007.
- Conclusion: Since p ≤ 0.05, reject H₀; there is strong evidence that the mean fill volume differs from 500 mL.
Comparing Z-Test Inputs and When to Choose Alternatives
| Scenario | Population Parameter Known? | Recommended TI-84 Test | Sample Size Guidance |
|---|---|---|---|
| Mean hypothesis, σ known | Yes | Z-Test | Use Z-test if n ≥ 30 or population normal |
| Mean hypothesis, σ unknown, use s | No | T-Test | Preferred for small n when σ is estimated by s |
| Proportion hypothesis (p) | Yes | 1-PropZTest | Requires np0 ≥ 10 and n(1−p0) ≥ 10 |
Common Pitfalls and Best Practices
- Do not use Z-test when σ is unknown and estimated by s. Use a T-test instead to account for extra uncertainty.
- Check randomness and independence. The validity of the Z-test depends on a random sample and, for without-replacement sampling, that the sample size is no more than 10% of the population.
- Beware of misentered σ. Make sure you enter the population standard deviation, not the sample standard deviation.
- Interpret p-values correctly. A non-significant result does not prove H₀; it indicates insufficient evidence to reject H₀ at the chosen α.
- Report effect size and context. Include the observed mean difference, z-statistic, p-value, and a practical conclusion tied to the problem context.
Reporting and Output Checklist for Your TI-84
After you run the Z-test, copy these items into your work or report for completeness: test name (Z-Test), alternative hypothesis direction (e.g., two-tailed), z-statistic (rounded to two or three decimals), degrees of freedom (not applicable for Z-test), p-value (to three or four decimals if small), sample mean x̄, sample size n, significance level α, and your decision (Reject H₀ or Fail to Reject H₀). Provide a concluding sentence that connects the decision to the real-world context.
Frequently Asked Questions About Z-Tests on TI-84
Can I use the Z-test for small samples?
Only if the underlying population is approximately normal and σ is known. For small samples with unknown σ, use a t-test.
What if I only have the sample standard deviation s?
Use a T-Test. The Z-test requires the population standard deviation σ.
How do I perform a one-tailed test?
When you select Z-Test, choose the appropriate Tail ( <μ0 for left-tailed, >μ0 for right-tailed, or ≠μ0 for two-tailed) before calculating.
Does the Z-test work for paired data?
No. For paired data, use a t-test for paired samples (T-Test with the Draws or list subtraction approach).
What is the difference between 1-PropZTest and Z-Test?
1-PropZTest is for a population proportion with known n, x, and p; Z-Test is for a population mean with known σ.
Assumptions and Limitations of the Z-Test
- Simple random sampling (or random assignment) is required.
- Either the population is normally distributed or n ≥ 30 for CLT-based approximation.
- The population standard deviation σ must be known; otherwise, use a t-test.
- For proportions, the success-failure condition np0 ≥ 10 and n(1−p0) ≥ 10 should hold.
Quick Tips for Using the TI-84 Z-Test Efficiently
- Store your sample data in L1 if you prefer Data entry; it makes reuse easier.
- Memorize common critical z-values: ±1.96 for two-tailed α = 0.05, ±1.645 for one-tailed α = 0.05.
- Use past answers (Ans) to recycle prior z or p when comparing multiple tests.
- Sketch the standard normal curve and shade the rejection region to visualize the p-value.
- Double-check that σ matches the population, not the sample, especially after copying data.
TI-84 Workflow Summary for Z-Test
Access the test, choose Stats or Data, enter known values, select the correct Tail, calculate, and interpret z and p. Compare p to α, decide on H₀, and write a contextual conclusion. When in doubt between Z-test and T-test, default to T-test if σ is estimated by s.
Tags: ti-84, z-test, statistics