Why Confidence Level Changes Interval Width
In “Interpreting the Confidence Level for a Mean,” you learned that a confidence level describes the long-run capture rate of an interval procedure. This tutorial looks at a related question: if we use the same sample data, what changes when we make a 90%, 95%, or 99% one-sample t interval?
The sample mean, sample standard deviation, sample size, degrees of freedom, and estimated standard error are all unchanged. The part that changes is the t critical value, \(t^*\). A higher confidence level uses a larger \(t^*\), which increases the margin of error and makes the interval wider. The interval remains centered at the same sample mean.
This is a tradeoff, not a flaw in the method. A narrower interval is more precise in the sense that it covers a smaller range of possible values. But, when all other aspects of the procedure stay fixed, a narrower interval has a lower confidence level. A wider interval offers a higher long-run chance of capturing the true mean, at the cost of being less precise.
Where the Width Comes From
Recall from “The Form of a One-Sample t Interval” that the interval is \(\bar{x}\pm t^*(s/\sqrt{n})\). The estimated standard error, \(s/\sqrt{n}\), measures the estimated variability of sample means. The margin of error is the distance from the center to either endpoint. The full width is the distance from the lower endpoint to the upper endpoint.
For a two-sided interval with confidence level \(C\), \(t^*\) is chosen to leave a total area of \(1-C\) in the two tails of the t distribution, with half in each tail. For example, a 90% interval leaves 10% outside the central area, or 5% in each tail. A 99% interval leaves only 1% outside the central area, or 0.5% in each tail. To leave less area in the tails, the positive cutoff \(t^*\) must move farther from zero.
If the data and degrees of freedom are fixed, that larger \(t^*\) directly increases the margin of error. It does not change \(\bar{x}\) or \(s/\sqrt{n}\). As a result, the lower endpoint moves down and the upper endpoint moves up by the same amount: the center stays at \(\bar{x}\), while the interval expands on both sides.
This comparison assumes the same data, the same one-sample t procedure, and the same degrees of freedom. Comparing intervals from different samples can be more complicated: the sample means, standard deviations, or sample sizes might differ as well as the confidence levels.
Worked Examples: Comparing Confidence Levels
Worked Example: Three Intervals From One Sample
A fictional random sample of 16 repair jobs has a mean completion time of 72 minutes and a standard deviation of 8 minutes. The sample was drawn from a population of 400 jobs. A plot of the sample values shows no strong skewness or outliers. Compare 90%, 95%, and 99% one-sample t intervals for the true mean repair time.
State. The parameter is the true mean repair time, \(\mu\), for the population of jobs, measured in minutes. We will compare three intervals made from the same sample.
Plan. As in “Building a One-Sample t Interval by Hand,” check the conditions for a t interval. The sample is described as random. The 10% condition is met because \(16<0.10(400)=40\), supporting independence when sampling without replacement. With \(n=16\), the plot’s lack of strong skewness or outliers supports using a t procedure. Thus, the one-sample t interval is appropriate.
Do. The degrees of freedom are \(df=n-1=15\). The estimated standard error is the same for all three intervals:
Using a t table or invT, the critical values for \(df=15\) are approximately 1.753 for 90%, 2.131 for 95%, and 2.947 for 99%. For instance, the 95% critical value uses left cumulative area \((1+0.95)/2=0.975\). The margin of error and endpoints for each interval are:
| Confidence level | \(t^*\) | Margin of error | Interval (minutes) | Width (minutes) |
|---|---|---|---|---|
| 90% | 1.753 | \(1.753(2)=3.506\) | \((72-3.506,\ 72+3.506)=(68.494,\ 75.506)\) | 7.012 |
| 95% | 2.131 | \(2.131(2)=4.262\) | \((72-4.262,\ 72+4.262)=(67.738,\ 76.262)\) | 8.524 |
| 99% | 2.947 | \(2.947(2)=5.894\) | \((72-5.894,\ 72+5.894)=(66.106,\ 77.894)\) | 11.788 |
The widths can also be checked by doubling each margin of error: \(2(3.506)=7.012\), \(2(4.262)=8.524\), and \(2(5.894)=11.788\) minutes.
Conclude. All three intervals are centered at 72 minutes because they use the same sample mean. The 99% interval is widest, and the 90% interval is narrowest. We are 99% confident that the true mean repair time is between 66.106 and 77.894 minutes; that interval is wider than the 90% interval because the 99% procedure is designed to capture the true mean more often in repeated appropriate samples.
Worked Example: Comparing the Cost of 99% Confidence
A fictional random sample of 25 residents gives a mean of 18.4 minutes for time spent traveling to a community service site, with a sample standard deviation of 6 minutes. Assume the population contains more than 250 residents, and a plot of the sample is reasonably symmetric with no outliers. Compare the widths of the 90%, 95%, and 99% intervals for the population mean travel time.
State. We want to see how much wider the 99% interval is than the 90% interval when both use the same sample.
Plan. The sample is random, and the 10% condition is met because \(25<0.10N\) when \(N>250\). The sample plot is reasonably symmetric with no outliers, so a one-sample t procedure is appropriate. Find the common estimated standard error, then compare widths using \(2t^*(s/\sqrt{n})\).
Do. Here \(df=25-1=24\), and the estimated standard error is
For \(df=24\), the approximate critical values are 1.711 for 90%, 2.064 for 95%, and 2.797 for 99%. The widths are:
The increase in width from 90% to 99% is approximately \(6.713-4.106=2.607\) minutes. Relative to the 90% width, this is about \(2.607/4.106\approx0.635\), or 63.5% wider.
Conclude. The 99% interval is about 63.5% wider than the 90% interval for this sample. This does not mean the sample became more variable: the data and standard error were unchanged. The extra width comes from choosing a larger t critical value to obtain a higher confidence level.
Worked Example: Choosing a Confidence Level Under a Width Limit
A fictional random sample of 36 water bottles has a mean fill volume of 502 milliliters and a standard deviation of 3 milliliters. The sample was randomly selected from a production lot of 500 bottles, and a plot shows no severe skewness or outliers. A report must use at least 95% confidence, and the interval must be no more than 2.2 milliliters wide. Which of the 95% and 99% intervals meets the width requirement?
State. We need to identify which eligible confidence level gives an interval no wider than 2.2 milliliters.
Plan. The random sample condition is met. The 10% condition holds because \(36<0.10(500)=50\). Since \(n=36\) and the plot has no severe skewness or outliers, a one-sample t procedure is appropriate. The requirement excludes 90% confidence, so compare the widths of the 95% and 99% intervals.
Do. The degrees of freedom are \(df=35\), and the estimated standard error is
For \(df=35\), the approximate critical values are 2.030 for 95% and 2.724 for 99%. The widths are:
The 95% interval has width 2.030 milliliters, which is below 2.2. The 99% interval has width 2.724 milliliters, which exceeds 2.2.
Conclude. The 95% interval meets both requirements: it is at least 95% confident and no more than 2.2 milliliters wide. The 99% interval meets the confidence requirement but not the width requirement. If the report instead required 99% confidence, these same data would not produce an interval satisfying the stated width limit.
Common Mistakes and AP Exam Tips
- Thinking a higher confidence level makes the interval narrower. Higher confidence requires a larger \(t^*\), so the margin of error and interval width increase. State the direction clearly: higher confidence means wider interval, when the data are held fixed.
- Changing the center when only confidence level changes. The center remains \(\bar{x}\). If two intervals based on the same data have different centers, something else changed or there is a calculation error.
- Comparing margins of error as if they were full widths. The margin of error is the distance from the center to one endpoint. The full width is twice the margin of error.
- Claiming that a 99% interval is more accurate because it is wider. A higher confidence level means a higher long-run capture rate, not a guarantee that a particular interval is closer to \(\mu\). A wider interval is less precise as a range, even though its procedure has higher confidence.
- Using confidence level to describe the data. A 99% interval does not mean that 99% of individual measurements fall between its endpoints. It estimates the population mean, and its confidence level describes the repeated-sampling procedure.
- Forgetting which factors are fixed in the comparison. The simple width comparison applies when the same data, sample size, and procedure are used. Across different samples, changes in \(s\), \(n\), or other sample features can also affect width.
For full-credit AP communication, connect the numerical result to the tradeoff and the context. For example: “Using the same sample, the 99% interval is wider than the 95% interval because its larger \(t^*\) increases the margin of error. The 99% procedure has a higher long-run capture rate, while the 95% interval gives a narrower estimate for the mean fill volume.”
Check Your Understanding
Use the relationship between confidence level and interval width to answer each question.
- For the same sample, a 90% interval has width 6 units. If the 95% interval is calculated from the same data, should its width be smaller, equal, or larger? Explain why.
- A student says, “The 99% interval is centered farther from the sample mean than the 90% interval.” Identify the error.
- A sample has an estimated standard error of 1.5 units. For a confidence level with \(t^*=2.0\), calculate the margin of error and the full interval width.
- Why does increasing the confidence level not change the estimated standard error when the same sample is used?
- A report requires at least 95% confidence and a width no greater than 5 units. The 95% interval is 4.6 units wide, and the 99% interval is 5.8 units wide. Which interval meets both requirements?