Lab 3: Confidence intervals

With solutions

Author

Your Name Here

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# load the dataset — 100 births recorded in North Carolina in 2004
load('data/ncbirths.RData')

A confidence interval gives a plausible range of values for a population parameter. The general formula is:

\[\bar{x} \pm c \times SE(\bar{x})\]

where \(c\) is a critical value that controls coverage: how often the interval captures the population mean. The empirical rule gives us whole-number critical values — in particular, \(c = 2\) yields approximately 95% coverage.

Computing a confidence interval by hand

How old are mothers at the time of birth in NC?

From Lab 2, we know how to get the point estimate and standard error from t.test(). We can then plug those directly into the interval formula.

# extract mother's age
mage <- ncbirths$mother.age

# run t.test
mage.tt <- t.test(mage)

# retrieve sample mean and standard error
mage.tt$estimate
mean of x 
     26.9 
mage.tt$stderr
[1] 0.6342895

Calculating by hand:

\[ 26.9 \pm 2\times0.63 = (26.3, 28.8) \]

With 95% confidence, the mean age of mothers at birth in NC in 2004 is estimated to be between 26.3 and 28.8 years.

CautionYour turn 1 [L4]

What is a plausible range for the mean birth weight of NC babies?

Extract birth.weight from ncbirths, run t.test(), and store the result as bweight.tt.

Then follow the example above to construct a 95% confidence interval using the empirical rule (\(c = 2\)).

Write a one-sentence interpretation.

# extract birth weight
bweight <- ncbirths$birth.weight

# run t.test
bweight.tt <- t.test(bweight)

# retrieve sample mean and standard error
bweight.tt$estimate
mean of x 
   7.1795 
bweight.tt$stderr
[1] 0.1434152

With 95% confidence, the mean birth weight of NC babies in 2004 is estimated to be between 6.89 and 7.47 lbs.

Confidence intervals from t.test()

Is there a faster way?

t.test() computes the confidence interval for you. The $conf.int component contains the lower and upper bounds directly.

# t.test computes the interval automatically — no arithmetic needed
mage.ci <- mage.tt$conf.int
mage.ci
[1] 25.64143 28.15857
attr(,"conf.level")
[1] 0.95

This matches what we calculated by hand above (with a slight difference because t.test() uses an exact \(t\) critical value rather than the approximation \(c = 2\)).

CautionYour turn 2 [L4]

What is a plausible range for the mean birth weight of NC babies?

Repeat your calculation from Your turn 1, but this time use t.test()$conf.int instead of the manual formula. Store the result as bweight.ci.

# same question as YT1, but letting t.test() do the arithmetic
bweight.ci <- bweight.tt$conf.int
bweight.ci
[1] 6.894933 7.464067
attr(,"conf.level")
[1] 0.95

The interval is nearly identical to the one from Your turn 1 — the small difference comes from t.test() using an exact \(t\) critical value rather than the \(c = 2\) approximation.

Adjusting coverage

What if we want to be more — or less — confident that the interval captures the true mean?

The conf.level argument controls coverage. The default is 0.95.

# 90% confidence interval for mother's age
t.test(mage, conf.level = 0.90)$conf.int
[1] 25.84683 27.95317
attr(,"conf.level")
[1] 0.9
# 99% confidence interval for mother's age
t.test(mage, conf.level = 0.99)$conf.int
[1] 25.2341 28.5659
attr(,"conf.level")
[1] 0.99

Higher coverage means a wider interval: you’re more confident it contains the true mean, but it pins down the mean less precisely.

CautionYour turn 3 [L4]

How long is a typical pregnancy in NC?

Extract weeks from ncbirths, run t.test(), and store the result as bweeks.tt. Then adjust conf.level to compare 90% and 99% intervals — no need to store these.

Which interval is narrowest?

# extract gestational age and run t.test (default 95% coverage)
bweeks <- ncbirths$weeks
bweeks.tt <- t.test(bweeks)

# adjust conf.level to compare widths — no need to store these
t.test(bweeks, conf.level = 0.90)$conf.int
[1] 38.07251 39.02749
attr(,"conf.level")
[1] 0.9
t.test(bweeks, conf.level = 0.99)$conf.int
[1] 37.7947 39.3053
attr(,"conf.level")
[1] 0.99

The 90% interval is narrowest.

Why not just use 100% confidence?

If higher coverage is better, why not always set conf.level = 1?

CautionYour turn 4 [L4]

Construct a 100% confidence interval for the population mean gestational age (number of weeks at birth).

a. Does the result make sense conceptually?

b. Is the result useful?

# use conf.level = 1
t.test(bweeks, conf.level = 1)$conf.int
[1] -Inf  Inf
attr(,"conf.level")
[1] 1
  1. Yes. The interval says we are certain that the mean is somewhere between \(-\infty\) and \(+\infty\).
  2. An infinitely wide interval provides no useful information about where the true mean lies. In practice, we accept a small chance of being wrong in exchange for a meaningful, finite range.
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