10	Basics of Confidence Intervals

10.1 Introduction to Estimation

The prior chapter introduced the form of statistical inference known as hypothesis testing. This chapter introduces the other common form of statistical inference: estimation. There are two forms of estimation: point estimation and interval estimation. Point estimation provides a single estimate of the parameter; interval estimation provides a range of values (confidence interval) that seeks to capture the parameter.

Figure 10.1 is a schematic of a confidence interval. The interval extends a margin of error above and below the point estimate. We may think of the margin of error as the “wiggle room” surrounding the point estimate. Therefore, a general formula for a confidence interval is:

FIGURE 10.1 Schematic of a confidence interval.

ILLUSTRATIVE EXAMPLE

Point and interval estimation (Body weight, 20–29-year-old U.S. men, 2000). The National Health and Nutrition Examination Survey (NHANES)^a assesses the health and nutritional status of adults and children in the United States using data from interviews and direct physical examinations. From 1999 to 2002, 712 men between the ages of 20 and 29 were examined. The mean body weight in this sample () was 183.0 pounds.^b This is the point estimate for the mean body weight in the population of men in this age group. The margin of error associated with this estimate for 95% confidence was ±3 pounds. Therefore, the 95% confidence interval for μ is 183 pounds ± 3 pounds = 180 to 186 pounds.

The confidence level of a confidence interval refers to the success rate of the method in capturing the parameter it seeks. For example, a 95% confidence interval for μ is constructed in such a way that 95% of like intervals will capture μ and 5% will fail to capture μ.

Figure 10.2 depicts five 95% confidence intervals for μ constructed from five independent SRSs of the same size from the same source population. It so happens that four of the five intervals captured μ in this schematic. (The third one missed.) In practice, we would not know which of the confidence intervals were successful and which were not. We would only know that, in the long run, 95% of the intervals will capture the parameter.

FIGURE 10.2 Five 95% confidence intervals from a sampling distribution of s.

10.2 Confidence Intervals for μ When σ Is Known

The Reasoning Behind Confidence Intervals

To understand how a confidence interval is constructed, we return to the notion of the sampling distribution of (Section 8.2). Figure 10.3 depicts repeated SRSs, each of n = 712, taken from the same population. Each independent sample calculates a mean () that contributes to a distribution of s. The value of population mean μ is not known, but the standard deviation of the population is known and is equal to 40.^c Based on the reasons discussed in Section 8.2, the sampling distribution of will be approximately Normal with mean μ and . (Recall that the is the standard deviation of the sampling distribution of .) Because of the 68–95–99.7 rule, we know that 95% of the s will fall within two standard errors of μ. In this instance, two standard errors = 2 × 1.5 pounds = 3 pounds. It follows that the interval ± 3.0 will capture the value of population mean μ 95% of the time. For the current example, a sample mean of 183 generates a 95% confidence for μ of (183 − 3) to (183 + 3) = 180 to 186 pounds.

FIGURE 10.3 Schematic of repeated independent samples of size n = 712 from the same source population with confidence intervals calculated with each sample mean.

In general, the confidence interval for μ is given by:

where m is the margin of error. For 95% confidence, m ≈ 2 × the standard error of the mean.

ILLUSTRATIVE EXAMPLE

95% confidence intervals for μ, σ known (Body weight, 20–29-year-old males). We wish to estimate the mean weight of a population in which body weight has a Normal distribution with σ = 40 pounds. Three independent SRSs each of n = 712 are selected from the population. These samples derive = 183, = 180, and = 184. Calculate 95% confidence intervals for μ based on each of these samples.

In each sample, pounds and m ≈ 2·SE = 2·1.5 = 3 pounds.

•   For sample 1, the confidence interval is ± m = 183 ± 3 = 180 to 186 (pounds).

•   For sample 2, the confidence interval is ± m = 180 ± 3 = 177 to 183 (pounds).

•   For sample 3, the confidence interval is ± m = 184 ± 3 = 181 to 187 (pounds).

Each of these confidence intervals has a 95% chance of capturing the population’s true mean weight μ.

Other Levels of Confidence

To calculate a (1 − α)100% confidence for μ, use this formula:

where z_1−α/2 represents a Standard Normal value with cumulative probability 1 − α/2 and . In this formula, α represents the probability a confidence interval will fail to capture μ. Common levels of α and confidence are:

	α-level	1 − α (confidence level)	z1−(α/2)
	0.10	90%	z1−(.10/2) = 1.645
	0.05	95%	1.960
	0.01	99%	2.576

Figure 10.4 depicts these levels of confidence on standard Normal curves.

FIGURE 10.4 z values for various levels of confidence; curves represent sampling distributions of means.

ILLUSTRATIVE EXAMPLE

Confidence intervals, various levels of confidence (Body weight, 20–29-year-old U.S. males). We select an SRS of n = 712 from a population of 20–29-year-old men and calculate = 183 pounds. The population standard deviation (σ) is 40. We propose to calculate 90%, 95%, and 99% confidence intervals for μ based on this information.

In each instance, .

•   The 90% confidence interval for μ is ± (z_1−.10/2)() = 183.0 ± (1.645)(1.5) = 183.0 ± 2.5 = 180.5 to 185.5 (pounds).

•   The 95% confidence interval for μ is ± (z_1−.05/2) () = 183.0 ± (1.960)(1.5) = 183.0 ± 2.9 = 180.1 to 185.9 (pounds).

•   The 99% confidence interval for μ is ± (z_1−.01/2) () = 183.0 ± (2.576)(1.5) = 183.0 ± 3.9 = 179.1 to 186.9 (pounds).

Thus, we can be 90% confident that the true mean weight in the population is between 180.5 and 185.5 pounds, we can be 95% confident that it is between 180.1 and 185.9 pounds, and we can be 99% confident that it is between 179.1 and 186.9 pounds. Notice that we “pay” for each additional step up in the confidence level with a broader confidence interval.

Notes

1.   Conditions for valid inference. This procedure assumes data were collected by an SRS (sampling independence), population standard deviation σ is known before data are collected, and the sampling distribution of is approximately Normal.^d

2.   Confidence level and confidence interval width. Figure 10.5 plots the 90%, 95%, and 99% confidence intervals for the illustrative example. Higher levels of confidence are associated with wider intervals.

3.   What confidence means. The confidence level of an interval tells you how often the method will succeed in capturing μ in the long run. With repeating samples, (1 − α)100% of the intervals will capture μ and (α)100% will not.

4.   Confidence intervals address random error only. Confidence intervals do not control for nonrandom sources of error, such as those that might be due to biased sampling, poor-quality information, and confounding. The method addresses random sampling error only.

5.   Table C. The last line of Appendix Table C lists z_1−α/2 critical values for various levels of confidence, thus providing a handy way to look up z values for confidence intervals with four significant-digit accuracy.

6.   Interpretation. Keep in mind that confidence intervals seek to capture population mean μ, not sample mean .

FIGURE 10.5 Higher confidence requires a wider interval.

Exercises

10.1    Misinterpreting a confidence interval. A pharmacist reads that a 95% confidence interval for the average price of a particular prescription drug is $30.50 to $35.50. Asked to explain the meaning of this, the pharmacist says “95% of all pharmacies sell the drug for between $30.50 and $35.50.” Is the pharmacist correct? Explain your response.

10.2    Newborn weight. A study reports that the mean birth weight of 81 full-term infants is 6.1 pounds. The study also reports “SE = 0.22 pounds.”

(a)   What is the margin of error of this estimate for 95% confidence?

(b)   What is the 95% confidence interval for μ?

(c)   What does it mean when we say that we have 95% confidence in the interval?

(d)   Recall that . Rearrange this equation to solve for σ. What was the standard deviation of birth weights in this population?

10.3    Newborn weight. A study takes an SRS from a population of full-term infants. The standard deviation of birth weights in this population is 2 pounds. Calculate 95% confidence intervals for μ for samples in which:

(a)   n = 81 and = 6.1 pounds

(b)   n = 36 and = 7.0 pounds

(c)   n = 9 and = 5.8 pounds

10.4    90% confidence intervals. Calculate 90% confidence intervals for μ based on the information reported in Exercise 10.3a–c.

10.5    SIDS. A sample of 49 sudden infant death syndrome (SIDS) cases had a mean birth weight of 2998 g. Based on other births in the county, we will assume σ = 800 g. Calculate the 95% confidence interval for the mean birth weight of SIDS cases in the county. Interpret your results.

10.6    99% confidence interval. Use the information in Exercise 10.5 to calculate a 99% confidence interval for μ.

10.3 Sample Size Requirements

The length of a confidence interval is equal to its upper confidence limit (UCL) minus its lower confidence limit (LCL).

Confidence interval length reflects the precision of the estimate. Narrow intervals reflect precision; wide intervals reflect imprecision.

The margin of error is also a reflection of the estimate’s precision. Margin of error m is equal to

where z_1–α/2 is a Standard Normal deviate with cumulative probability 1−α/2, σ is the population standard deviation, and n is the sample size. This formula is rearranged to determine the sample size needed to achieve margin of error m:

Results are rounded up to ensure that we achieve the stated precision.

ILLUSTRATIVE EXAMPLE

Sample size requirement (Body weight, 20–29-year-old males). We have established that the body weights of 20–29-year-old U.S. males are σ = 40 pounds. How many observations do we need in order to estimate mean body weight μ with 95% confidence and a margin of error of plus or minus 10 pounds?

Solution: Recall that for 95% confidence, (from Appendix Table B). Therefore, the required sample size is . Round this up to 62.

What size sample is needed to estimate μ with margin of error plus or minus 5 pounds?

Solution: . Round this to 246.

What size sample is needed to cut the margin down to 1 pound?

Solution: . Round this up to 6147.

The formula lets us know that three factors contribute to the margin of error. These are:

1.    which determines the confidence level of the interval. Decreasing confidence will shrink and the margin of error.

2.   σ, which is the standard deviation of the variable. A variable with high variability will obscure population mean μ.

3.   n, which is the sample size. Increasing the sample size decreases the margin of error according to the square root of n. For example, quadrupling the sample size will cut the margin of error in half.

Exercises

10.7    Hemoglobin. Hemoglobin levels in 11-year-old boys vary according to a Normal distribution with σ = 1.2 g/dL.

(a)   How large a sample is needed to estimate mean μ with 95% confidence so the margin of error is no greater than 0.5 g/dL?

(b)   How large a sample is needed to estimate μ with margin of error 0.5 g/dL with 99% confidence?

10.8    Sugar consumption. Based on prior studies, a dental researcher is willing to assume that the standard deviation of the weekly sugar consumption in children in a particular community is 100 g.

(a)   How large a sample is needed to estimate mean sugar consumption in the community with a margin of error 10 g at 95% confidence?

(b)   How many kids should be studied if the researcher is willing to accept a margin of error of 25 g at 95% confidence?

10.4 Relationship Between Hypothesis Testing and Confidence Intervals

You can use a (1 − α)100% confidence interval for μ to predict whether a two-sided test of H₀: μ = μ₀ will be significant at the α-level of significance. When the value of the parameter identified in the null hypothesis (μ₀) falls outside the interval, the results will be statistically significant (reject H₀). When the value of the parameter identified in the null hypothesis is captured by the interval, the results will not be statistically significant (retain H₀).

ILLUSTRATIVE EXAMPLE

Significance test from confidence interval. A prior illustrative example calculated confidence intervals for the mean body weight of the U.S. 20–29-year-old male population. We use this information to test H₀: μ = 180 pounds at differing α levels.

•   The 90% confidence interval for μ was 180.5 to 185.5 (pounds). In testing H₀: μ = 180 pounds at α = 0.10, results are statistically significant (reject H₀).

•   The 95% confidence interval for μ was 180.1 to 185.9 pounds. In testing H₀: μ = 180 pounds at α = 0.05, these results are statistically significant (reject H₀).

•   The 99% confidence interval for μ was 179.1 to 186.9 pounds. In testing H₀: μ = 180 pounds at α = 0.01, these results are not statistically significant (retain H₀).

Figure 10.6 illustrates the 95% and 99% confidence intervals for this problem in relation to the sampling distribution of assuming H₀: μ = 180. Notice that the 95% confidence interval fails to capture μ₀, while the 99% confidence interval captures it.

FIGURE 10.6 Relation between confidence intervals and hypothesis test. (1 − α)100% confidence intervals that exclude μ₀ are significant at the α level of significance.

Exercise

10.9    P-value and confidence interval. A two-sided test of H₀: μ = 0 yields a P-value of 0.03. Will the 95% confidence interval for μ include 0 in its midst? Will the 99% confidence interval for μ include 0? Explain your reasoning in each instance.

Summary Points (Introduction to Confidence Intervals)

1.   This chapter introduces the second form of statistical inference: estimation.

2.   Sample mean is the point estimator of population parameter μ.

3.   Interval estimates are called confidence intervals. Confidence intervals can be calculated at almost any level of confidence. The most common levels of confidence are 90%, 95%, and 99%.

4.   The (1 − α)100% confidence interval has a (1 − α)100% chance of capturing the true value of the parameter and an α chance of not capturing the true value.

5.   A (1 − α)100% confidence interval for , where is the sample mean and is the 1 − α/2 percentile on a Standard Normal distribution.

(a)   This confidence interval consists of the point estimate and surrounding margin of error (m): ± m.

(b)   Margin of error m is a measure of the precision of the estimate.

(c)   The confidence interval is written as (LCL to UCL) or (LCL, UCL), where LCL represents the lower confidence limit and UCL represents the upper confidence limit.

6.   z confidence interval procedure for μ requires the following conditions:

(a)   Data are an SRS from the population (or reasonable approximation thereof).

(b)   The population is Normal or the sample is large enough to invoke the central limit theorem.

(c)   Population standard deviation σ is known before collecting data.

(d)   The data are accurate.

7.   The sample size requirement to achieve margin of error m when estimating .

(a)   Use an educated guess as standard deviation σ. If an educated guess for σ is not available, then first do a pilot study.

(b)   m is the margin of error you can “live with.” Smaller-and-smaller margins of error require larger-and-larger sample sizes.

(c)   Increasing the required level of confidence mandates larger-and-larger sample sizes.

Vocabulary

Confidence interval

Confidence interval length

Interval estimation

Level of confidence

Lower confidence limit

Margin of error

Point estimate

Point estimation

Upper confidence limit (UCL)

Review Questions

10.1    Fill in the blank: The two forms of estimation are point estimation and _____________ estimation.

10.2    While the goal of hypothesis testing is to test a claim, the goal of estimation is to estimate a _____________.

10.3    Select the best response: is a(n) _____________ estimator of μ because as a sample size gets larger and larger, we expect to get closer and closer to the true value of μ.

(a)   biased

(b)   reliable

(c)   unbiased

10.4    Select the best response: This is the “wiggle room” placed around the point estimate to derive a confidence interval.

(a)   standard deviation

(b)   standard error

(c)   margin of error

10.5    Select the best response: This is the larger of the two numbers listed when reporting a confidence interval.

(a)   point estimate

(b)   lower confidence limit

(c)   upper confidence limit

10.6    Select the best response: The confidence interval for the mean seeks to capture the _____________.

(a)   sample mean

(b)   population mean

(c)   population standard deviation

10.7    Select the best response: The standard error of the mean is inversely proportional to the _____________.

(a)   standard deviation

(b)   sample size

(c)   square root of the sample size

10.8    What percentage of 90% confidence intervals for μ will fail to capture μ?

10.9    What percentage of (1 − α)100% confidence intervals for μ will fail to capture μ?

10.10  Select the best response: Which of the following reflects the precision of as an estimate of μ?

(a)   the margin of error m

(b)   the confidence interval length

(c)   both (a) and (b)

10.11  Select the best response: The margin of error m for the (1 − α)100% confidence interval for μ is equal to z_1−α/2 times the _____________.

(a)   sample mean

(b)   standard deviation

(c)   standard error of the mean

10.12  Select the best response: To cut the margin of error in half, we must _____________ the sample size.

(a)   double

(b)   triple

(c)   quadruple

10.13  Select the best response: For the same set of data, which confidence interval will be the longest?

(a)   the 90% confidence interval for μ

(b)   the 95% confidence interval for μ

(c)   the 99% confidence interval for μ

10.14  Select the best response: Which of the following α levels is associated with 95% confidence?

(a)   1%

(b)   5%

(c)   10%

10.15  Select the best response: Which of the following z critical values is used to achieve 95% confidence?

(a)   z_0.90

(b)   z_0.95

(c)   z_0.975

Exercises

10.10  Laboratory scale. The manufacturer of a laboratory scale claims their scale is accurate to within 0.0015 g. You read the documentation for the scale and learn that this means that the standard deviation of an individual measurement (σ) is equal to 0.0015 g. Assume measurements vary according to a Normal distribution with μ equal to the actual weight of the object. You weigh the same specimen twice and get readings of 24.31 and 24.34 g. Based on this information, calculate a 95% confidence interval for the true weight of the object.

10.11  Antigen titer. A vaccine manufacturer analyzes a batch of product to check its titer. Immunologic analyses are imperfect, and repeated measurements on the same batch are expected to yield slightly different titers. Assume titer measurements vary according to a Normal distribution with mean μ and σ = 0.070. (The standard deviation is a characteristic of the assay.) Three measurements demonstrate titers of 7.40, 7.36, and 7.45. Calculate a 95% confidence interval for true concentration of the sample.

10.12  Newborn weight. The 95% confidence interval for the mean weight of infants born to mothers who smoke is 5.7 to 6.5 pounds. The mean weight for all newborns in this region is 7.2 pounds. Is the birth weight of the infants in this sample significantly different from that of the general population at α = 0.05? Explain your response.

10.13  Reverse engineering the confidence interval. The 95% confidence interval in Exercise 10.12 (5.7 to 6.5 pounds) was calculated with the usual formula Confidence intervals constructed in this way are symmetrical around the mean; is the mid-point of the confidence interval.

(a)   What is the value of the sample mean used to calculate the confidence interval?

(b)   What is the margin of error of the confidence interval?

(c)   What is the standard error of the estimate?

(d)   Calculate a 99% confidence interval for μ.

(e)   The sample mean is significantly different from 7.2 pounds at α = 0.05. Is the difference significant at α = 0.01?

10.14  True or false? Answer “true” or “false” in each instance. Explain each response.

(a)   95% of 95% confidence intervals for a mean will fail to capture .

(b)   95% of 95% confidence intervals for a mean will fail to capture μ.

(c)   95% of 95% confidence intervals for a mean will capture μ.

10.15  True or false? A confidence interval for μ is 13 ± 5.

(a)   The value 5 in this expression is the estimate’s standard error.

(b)   The value 13 in this expression is the estimate’s margin of error.

(c)   The value 5 in this expression is the estimate’s margin of error.

10.16  True or false?

(a)   Populations with high variability will produce estimates with large margins of error (all other things being equal).

(b)   Studies with small sample sizes will tend to have large margins of error.

10.17  Lab reagent, 90% confidence interval for true concentration. Exercise 9.20 presented six measurements of a reagent in which the true concentration of the reagent was uncertain. The sample mean based on these six measurements was 4.9883 mg/dL. The distribution of an infinite number of concentration measurements taken from the solution is assumed to be Normal with μ equal to the true concentration of the solution and σ = 0.2 mg/dL. Calculate a 90% confidence interval for the true concentration of the solution based on the current set of observations. What do you conclude?

10.18  Lab reagent, 99% confidence interval for true concentration. Calculate a 99% confidence interval for the true concentration of the solution considered in the prior exercise. Interpret the results. How does this confidence interval compare the 90% confidence interval?

______________

^a National Center for Health Statistics, Centers for Disease Control.

^b Ogden, C. L., Fryar, C. D., Carroll, M. D., & Flegal, K. M. (2004). Mean body weight, height, and body mass index, United States 1960–2002. Advance Data from Vital and Health Statistics, 347, 1–17. See Table 6. Values have been rounded for illustrative purposes.

^c Chapter 11 will eliminate the need to specify the value of the population standard deviation ahead of time.

^d See Section 9.5 for comments about the Normality assumption.