From the Hand Calculation to the Calculator
In Constructing a One-Proportion z-Interval by Hand, you calculated the sample proportion, estimated standard error, margin of error, and endpoints. The calculator’s 1-PropZInt command performs those calculations for you. To use it well, you still need to enter the correct information, check that the interval method is appropriate, and understand what the output represents.
The command needs three inputs: the number of successes \(x\), the sample size \(n\), and the confidence level. It uses those values to calculate the interval for the population proportion \(p\). The calculator does not decide whether your sample is random or whether the interval conditions are met; you must check those from the study description.
Entering the Values
On a TI-84, press STAT, move to TESTS, and choose 1-PropZInt. The entry screen has fields for \(x\), \(n\), and C-Level. Enter the success count in \(x\), the total number of observations in \(n\), and the confidence level as a decimal. For example, enter \(0.95\) for 95% confidence—not 95.
After entering the values, select Calculate. The output gives the lower and upper endpoints of the interval and typically also displays \(\hat{p}\) and \(n\). Calculator models and display settings can differ, so focus on what each number means rather than expecting a particular screen layout. The interval endpoints are proportions; multiply by 100 if you want to describe them as percentages.
- \(x\): the number of observations with the characteristic of interest.
- \(n\): the total number of observations, including successes and failures.
- C-Level: the confidence level as a decimal, such as \(0.90\), \(0.95\), or \(0.99\).
The command uses the same interval formula you used by hand:
Here \(z^*\) is the critical value for the selected confidence level. In Finding Critical Values \(z^*\) for Common Confidence Levels, you learned how to find that value. When checking calculator output, use the same confidence level and keep extra digits during the hand calculation. Small differences in the last displayed digit can come from rounding.
Check Conditions Before You Trust the Output
As in the earlier tutorials on the 10% condition and Large Counts condition, check whether the data support a one-proportion \(z\)-interval before interpreting the calculator’s result. A calculator can return endpoints even when the conditions are not met.
- Random: The data come from a random sample or a suitable random process.
- 10% condition: If sampling without replacement from a finite population of size \(N\), verify \(n\leq0.10N\).
- Large Counts: The observed numbers of successes and failures are each at least 10: \(x\geq10\) and \(n-x\geq10\).
The calculator’s output should match the hand-calculation method from the previous tutorial: calculate \(\hat{p}\), use it to find the estimated standard error, apply the critical value for the entered confidence level, and obtain the two endpoints. The worked examples show how to check each part.
Worked Example: A 95% Interval for Composting at Home
A community team wants to estimate the proportion of households in its area that compost food scraps at home. The team selects a random sample of 160 households from a population of 5,000 households. Of the sampled households, 64 compost food scraps. Use the calculator to find a 95% one-proportion \(z\)-interval and verify it by hand.
Let \(p\) be the proportion of all households in this area that compost food scraps at home. We will estimate \(p\) with a 95% confidence interval.
The problem states that the 160 households were selected randomly. Since the sample is taken without replacement, check the 10% condition: \(0.10(5000)=500\), and \(160\leq500\). There are 64 successes and \(160-64=96\) failures, both at least 10, so the Large Counts condition is met. A one-proportion \(z\)-interval is appropriate.
Enter \(x=64\), \(n=160\), and C-Level \(=0.95\) in 1-PropZInt, then select Calculate. Check the displayed interval against the hand calculation below.
Report the interval as an estimate of the proportion of all households in this area that compost food scraps at home.
Calculate the estimate and estimated standard error. The calculator’s displayed \(\hat{p}\) should be \(64/160=0.40\). Using that value:
For a 95% confidence level, \(z^*=1.96\). The margin of error is:
So the hand-calculated endpoints are:
The calculator should give an interval close to \((0.3241,\,0.4759)\), with \(\hat{p}=0.40\) and \(n=160\) if those values are included on its output screen. We are 95% confident that about 32.4% to 47.6% of all households in this area compost food scraps at home. The confidence level describes the long-run success rate of the interval method, not a 95% probability that this particular interval contains the fixed value of \(p\).
Match the Output to the Formula
When you compare the calculator output with your hand calculation, match values by their roles. The center of the interval should be \(\hat{p}=x/n\). The two endpoints should be approximately the center minus and plus the margin of error. If the confidence level increases while \(x\) and \(n\) stay fixed, the interval should become wider because the critical value is larger.
You do not need to reproduce every hidden calculator step in an answer that asks you to use technology. But you should be able to show enough work to verify the result: state the inputs, show \(\hat{p}\) and the interval, and explain why the conditions support the method. If asked to compare with the hand calculation, show the standard error and margin of error as well.
Worked Example: A 90% Interval for a School Garden
A school garden committee takes a random sample of 120 students from a school with 2,000 students. In the sample, 39 students say they helped with the garden during the current school year. Find a 90% confidence interval for the proportion of all students at the school who helped.
Check conditions. The sample is stated to be random. For sampling without replacement, \(0.10(2000)=200\), and \(120\leq200\), so the 10% condition is met. There are 39 successes and \(120-39=81\) failures, both at least 10.
On the calculator, enter \(x=39\), \(n=120\), and C-Level \(=0.90\). The sample proportion is:
To check the calculator, the estimated standard error and margin of error are:
The calculator’s endpoints should be close to:
Thus, the 90% interval is approximately \((0.255,\,0.395)\). We are 90% confident that between about 25.5% and 39.5% of all students at this school helped with the garden during the current school year. The calculator result should be close to the hand result; a difference in the final displayed digit is not a concern if it is caused by rounding.
Worked Example: A 99% Interval for a Reusable Cup Program
A campus sustainability group wants to estimate the proportion of students who bring a reusable cup to campus at least once a week. A random sample of 200 students is selected from a campus population of 6,000; 87 report bringing a reusable cup at least once a week. Use 1-PropZInt with a 99% confidence level and check the result by hand.
Check conditions. The sample is random. The 10% limit is \(0.10(6000)=600\), and \(200\leq600\). There are 87 successes and \(200-87=113\) failures, both at least 10.
Enter \(x=87\), \(n=200\), and C-Level \(=0.99\). The sample proportion is:
For a 99% interval, \(z^*\approx2.5758\). The estimated standard error and margin of error are:
The endpoints are approximately:
The calculator should display an interval close to \((0.3447,\,0.5253)\). We are 99% confident that about 34.5% to 52.5% of all students on this campus bring a reusable cup to campus at least once a week. Compared with a lower confidence level using the same sample, this 99% interval is wider.
Common Mistakes and AP Exam Tips
- Entering the confidence level as a percent. Enter \(0.95\) for 95% confidence, not 95. A value outside the intended range may lead to an error or an unintended result.
- Reversing \(x\) and \(n\). Enter the success count for \(x\) and the total sample size for \(n\). The success count cannot exceed the total.
- Entering a percentage instead of a count. If 39 of 120 students are successes, enter \(x=39\), not \(0.325\). The command calculates \(\hat{p}\) from the counts.
- Choosing the wrong definition of success. Decide what characteristic counts as a success before entering \(x\). The count, sample proportion, and contextual conclusion must all refer to that same characteristic.
- Assuming calculator output verifies the conditions. It does not. State the random-sample evidence, check the 10% condition when needed, and verify that both observed counts are at least 10.
- Reporting only the screen numbers. A strong AP response identifies the population proportion, gives the interval in context, and uses the requested confidence level. Include the units or explain that the endpoints are proportions or percentages.
- Worrying about a tiny rounding difference. If the calculator and hand endpoints differ slightly, check that you used the same confidence level and did not round intermediate values too soon. The results should agree apart from rounding.
Key Takeaway
For a one-proportion confidence interval, enter the number of successes \(x\), sample size \(n\), and confidence level as a decimal in 1-PropZInt. The output should match the interval calculated by hand using \(\hat{p}=x/n\) and \(\hat{p}\pm z^*\sqrt{\hat{p}(1-\hat{p})/n}\). Always check the conditions and explain what the interval estimates.
Check Your Understanding
For each question, show how you would enter or check the values and explain the result where requested.
- A random sample has 52 successes among 130 observations. What values should be entered for \(x\) and \(n\) in 1-PropZInt?
- What should you enter in the C-Level field for an 85% confidence interval?
- A calculator gives an interval for \(x=42\), \(n=140\), and C-Level \(=0.95\). What sample proportion should appear in the output?
- A random sample of 150 people is drawn without replacement from a population of 1,200. Does the sample meet the 10% condition? Show the comparison.
- Why should you still check the random, 10%, and Large Counts conditions even when 1-PropZInt produces an interval?