9.4. Confidence Bounds for the Poulation Mean When σ is Known

In the previous chapter, we developed two-sided confidence intervals to quantify uncertainty in our estimates of population parameters. However, there are many practical situations where our interest lies primarily in one direction: establishing that a parameter exceeds a minimum threshold or does not exceed a maximum limit. In such cases, one-sided confidence bounds provide a more appropriate and often more precise approach.

While confidence intervals provide upper and lower limits for a parameter value, confidence bounds establish a threshold in only one direction. The logic and derivation follow the same principles as two-sided intervals, but with an important difference in how we allocate the probability of error.

Road Map 🧭

Fill Content

9.4.1. Why One‑Sided Bounds?

Two-sided confidence intervals capture the unknown population mean μ on both sides of our point estimate \(\bar{x}\). However, in many practical applications, scientific questions have a directional nature that calls for a directional guarantee:

• Safety assessments: The mean toxin level must not exceed a legal limit.

• Quality control: The average tensile strength must be at least a promised specification.

• Engineering reliability: A component’s mean time to failure must be greater than a minimum benchmark.

• Performance standards: A drug’s efficacy must be above a specified threshold.

In these situations, a one-sided confidence bound is not only more directly aligned with the research question but is also typically narrower than a two-sided interval at the same confidence level, providing a more precise answer to the specific question of interest.

9.4.2. Recap of the Two‑Sided Formula

Before developing one-sided bounds, let’s briefly review the two-sided interval. When σ is known, the \(100(1-\alpha)\%\) two-sided confidence interval is:

\[\bar{x}\;\pm\;z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\]

For a 95% confidence interval (α = 0.05), we use \(z_{\alpha/2} = z_{0.025} \approx 1.96\).

In this formula, the critical value \(z_{\alpha/2}\) places probability α/2 in each tail of the distribution, reflecting our equal concern about errors in either direction.

9.4.3. Deriving the Upper Confidence Bound

For a one-sided bound, we’re concerned only about error in one direction. We start with the same pivotal quantity from Chapter 9.3:

\[Z = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}} \sim \text{Normal}(0,1)\]

For an upper confidence bound, we want to state that the parameter μ is below some calculated value with high probability. This means we place all the error probability α in the upper tail:

\[P\left(Z \leq z_{\alpha}\right) = 1-\alpha\]

Where \(z_{\alpha}\) is the critical value with area α to its right under the standard normal curve.

Fig. 9.7 Figure 9.7: For an upper confidence bound, all error probability α is placed in one tail.

Now we solve for μ by algebraically manipulating the inequality:

1. Replace Z with its formula:

   \[P\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}} \leq z_{\alpha}\right) = 1-\alpha\]

2. Multiply both sides by \(\sigma/\sqrt{n}\):

   \[P\left(\bar{X}-\mu \leq z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\right) = 1-\alpha\]

3. Subtract \(\bar{X}\) from both sides and multiply by -1 (which reverses the inequality):

   \[P\left(\mu \leq \bar{X} + z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\right) = 1-\alpha\]

4. This gives us our probability statement. When we observe a specific sample with mean \(\bar{x}\), we obtain the \(100(1-\alpha)\%\) upper confidence bound (UCB):

   \[\text{UCB}: \mu \leq \bar{x} + z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\]

   Or written as an interval:

   \[\left(-\infty,\; \bar{x} + z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\right]\]

This bound states that we are \(100(1-\alpha)\%\) confident that the true population mean does not exceed the calculated upper bound.

9.4.4. Deriving the Lower Confidence Bound

By similar reasoning, we can derive a lower confidence bound (LCB). For a lower bound, we want to state that the parameter μ is above some calculated value with high probability.

We place all the error probability α in the lower tail:

\[P\left(Z \geq -z_{\alpha}\right) = 1-\alpha\]

Following the same algebraic steps, we arrive at:

\[P\left(\mu \geq \bar{X} - z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\right) = 1-\alpha\]

When we observe a specific sample mean \(\bar{x}\), this gives us the \(100(1-\alpha)\%\) lower confidence bound (LCB):

\[\text{LCB}: \mu \geq \bar{x} - z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\]

Or written as an interval:

\[\left[\bar{x} - z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}},\; \infty\right)\]

This bound states that we are \(100(1-\alpha)\%\) confident that the true population mean is at least as large as the calculated lower bound.

Connection Between One- and Two-Sided Procedures

An interesting relationship exists between one- and two-sided procedures:

• A 95% two-sided confidence interval uses \(z_{0.025} \approx 1.96\)

• A 90% one-sided confidence bound uses \(z_{0.10} \approx 1.28\)

• A 95% one-sided confidence bound uses \(z_{0.05} \approx 1.64\)

Notice that a 95% one-sided bound and a 90% two-sided interval use the same critical value. This is because in the two-sided case, we allocate α/2 to each tail, while in the one-sided case we allocate all α to a single tail.

This table summarizes key critical values:

Confidence Level	Two-sided \(z_{\alpha/2}\)	One-sided \(z_{\alpha}\)
90%	1.645	1.282
95%	1.960	1.645
99%	2.576	2.326

In R, we can find these critical values using:

# For a 95% one-sided bound (alpha = 0.05)
z_alpha <- qnorm(0.95)  # Note: 0.95 = 1 - 0.05
z_alpha  # Returns 1.645

# For a 95% two-sided interval (alpha = 0.05)
z_alpha_2 <- qnorm(0.975)  # Note: 0.975 = 1 - 0.05/2
z_alpha_2  # Returns 1.96

9.4.5. Choosing Between Bounds and Intervals

The choice between a one-sided bound and a two-sided interval should be made based on the scientific question, not on which approach gives more favorable results. Some guidelines:

• Choose before data collection: Decide whether a one- or two-sided procedure is appropriate before collecting or analyzing any data.

• Match to research question: Use one-sided bounds when the research question naturally has a directional component.

• Avoid post-hoc selection: One-sided bounds are not a license to cherry-pick whichever tail looks better after seeing the data. Such practice invalidates the stated confidence level.

• Regulatory context: In regulated industries (pharmaceuticals, environmental testing), regulatory agencies may have specific requirements regarding the use of one- versus two-sided procedures.

Statistical Integrity

Changing from a planned two-sided interval to a one-sided bound after seeing the data is a form of p-hacking that invalidates the stated confidence level. Always pre-specify your analysis approach.

Example 💡- Lead Content in Drinking Water

An environmental agency samples n = 40 water taps in an older neighborhood. Laboratory analysis reports a sample mean lead concentration of \(\bar{x} = 12.7\) ppb (parts per billion). Historical data suggest the population standard deviation is σ = 4.5 ppb. The Environmental Protection Agency (EPA) action level for lead in drinking water is 15 ppb.

The question of interest is: “Is the mean lead level in this neighborhood safely below the EPA action limit?” This directional question naturally calls for an upper confidence bound.

To construct a 95% upper confidence bound:

Step 1: Find the critical value \(z_{0.05}\)

\[z_{0.05} = 1.645\]

Step 2: Calculate the margin of error

\[\text{ME} = z_{0.05} \cdot \frac{\sigma}{\sqrt{n}} = 1.645 \cdot \frac{4.5}{\sqrt{40}} \approx 1.17 \text{ ppb}\]

Step 3: Determine the upper bound

\[\text{UCB} = \bar{x} + \text{ME} = 12.7 + 1.17 = 13.87 \text{ ppb}\]

Interpretation: We are 95% confident that the true mean lead level in the neighborhood does not exceed 13.87 ppb, which is below the EPA action limit of 15 ppb. This provides statistical evidence that the neighborhood’s water supply is in compliance with EPA standards.

Here’s the calculation in R:

# Input values
n <- 40
xbar <- 12.7  # sample mean (ppb)
sigma <- 4.5  # population standard deviation (ppb)
conf_level <- 0.95

# Calculate upper confidence bound
z_alpha <- qnorm(conf_level)
me <- z_alpha * sigma / sqrt(n)
ucb <- xbar + me

# Display result
cat("95% Upper Confidence Bound:", round(ucb, 2), "ppb\n")
# Output: 95% Upper Confidence Bound: 13.87 ppb

9.4.6. Sample Size Planning for One-Sided Bounds

Just as with two-sided intervals, we can determine the sample size needed to achieve a specified margin of error in a one-sided bound. For a target one-sided margin of error E, we need:

\[E = z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\]

Solving for n:

\[n \geq \left(\frac{z_{\alpha} \cdot \sigma}{E}\right)^2\]

Because we use \(z_{\alpha}\) rather than \(z_{\alpha/2}\), the required sample size for a one-sided bound with confidence level 1-α is smaller than for a two-sided interval with the same confidence level and margin of error.

For example, if we want a 95% upper confidence bound for the lead level with a margin of error of no more than 1 ppb:

# Target margin of error
E <- 1  # ppb

# Required sample size
n_required <- ceiling((z_alpha * sigma / E)^2)
n_required  # Result: 55

By comparison, a 95% two-sided interval with the same margin of error would require:

z_alpha_2 <- qnorm(0.975)
n_required_2sided <- ceiling((z_alpha_2 * sigma / E)^2)
n_required_2sided  # Result: 78

This shows that the one-sided bound requires approximately 30% fewer observations to achieve the same precision (in the direction of interest).

9.4.7. Assumptions and Cautions

The one-sided confidence bound procedure relies on the same assumptions as the two-sided confidence interval:

1. Simple Random Sample: The data must be a simple random sample from the target population.

2. Known σ: The population standard deviation is known, a simplifying assumption that we’ll relax in the next chapter.

3. Normality OR large n: Either the population follows a normal distribution, or the sample size is large enough for the Central Limit Theorem to ensure that \(\bar{X}\) is approximately normally distributed.

Several cautions apply when using one-sided bounds:

• Direction must be justified a priori: The choice of an upper or lower bound should be based on the research question, not on the observed data.

• Regulatory requirements: Some regulatory agencies may specifically require two-sided intervals rather than one-sided bounds.

• Non-resistant to outliers: Like the two-sided interval, one-sided bounds based on \(\bar{x}\) are sensitive to outliers.

• Heavy-tailed distributions: For severely skewed or heavy-tailed distributions, larger sample sizes may be needed for the Central Limit Theorem to apply adequately.

9.4.8. Bringing It All Together

In this section, we’ve continued with the simplifying assumption that the population standard deviation σ is known. In the next chapter, we’ll relax this assumption and develop procedures for when σ must be estimated from the sample data.

Key Takeaways 📝

• One-sided confidence bounds are appropriate when the research question involves a directional concern about a parameter.

• An upper confidence bound is \(\bar{x} + z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\), providing a value that the parameter likely does not exceed.

• A lower confidence bound is \(\bar{x} - z_{\alpha} \cdot \frac{\sigma}{\sqrt{n}}\), providing a value that the parameter likely exceeds.

• One-sided bounds use different critical values from two-sided intervals at the same confidence level.

• One-sided bounds require smaller sample sizes than two-sided intervals for the same confidence level and margin of error.

• The decision to use a one-sided bound should be made before data collection and analysis.

Exercises

1. A battery manufacturer wants to ensure that the mean battery life exceeds 40 hours. A sample of 25 batteries has a mean life of 42.3 hours. Assuming σ = 5 hours is known from extensive testing, construct a 95% lower confidence bound for the mean battery life. Does this provide evidence that the mean exceeds 40 hours?

2. Environmental regulations require that the mean concentration of a pollutant in factory discharge not exceed 3.5 ppm. A random sample of 36 discharge measurements yields \(\bar{x} = 3.2\) ppm. If σ = 0.8 ppm, construct a 99% upper confidence bound. Does the factory appear to be in compliance?

3. A quality control engineer wants to establish that the mean tensile strength of steel cables is at least 5000 N with a margin of error of 100 N at 95% confidence. If σ = 300 N, how many cables should be tested?

4. Explain why switching from a planned two-sided CI to a one-sided bound that “looks better” after seeing the data inflates the Type I error rate. Illustrate with a simple example.

5. A marketing claim states that a new energy drink provides at least 6 hours of improved alertness. A sample of 16 participants shows a mean alertness duration of 6.4 hours with a known standard deviation of 1.2 hours. Construct a 90% lower confidence bound and interpret it in context. How would the bound change if you used a 95% confidence level instead?