# 6.4: Using R to Find Confidence Intervals

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The confidence interval for a population’s mean, $$\mu$$, given an experimental mean, $$\bar{x}$$, for $$n$$ samples is defined as

$\mu = \bar{x} \pm \frac {z \sigma} {\sqrt{n}} \nonumber$

if we know the population's standard deviation, $$\sigma$$, and as

$\mu = \bar{x} \pm \frac {t s} {\sqrt{n}} \nonumber$

if we assume that the sample's standard deviation, $$s$$, is a reasonable predictor of the population's standard deviation. To find values for $$z$$ we use R'sqnorm()function, which takes the form

qnorm(p)

wherepis the probability on one side of the normal distribution curve that a result is not included within the confidence interval. For a 95% confidence interval, $$p = 0.05/2 = 0.025$$ because the total probability of 0.05 is equally divided between both sides of the normal distribution. To find $$t$$ we use R'sqt()function, which takes the form

qt(p, df)

wherepis defined as above and wheredfis the degrees of freedom or $$n - 1$$.

For example, if we have a mean of $$\bar{x} = 12$$ for 10 samples with a known standard deviation of $$\sigma = 2$$, then for the 95% confidence interval the value of $$z$$ and the resulting confidence interval are

# for a 95% confidence interval, alpha is 0.05 and the probability, p, on either end of the distribution is 0.025;

# the value of z is positive on one side of the normal distribution and negative on the other side;

# as we are interested in just the magnitude, not the sign, we use the abs() function to return the absolute value

z = qnorm(0.025) conf_int_pop = abs(z * 2/sqrt(10)) conf_int_pop

 1.23959

Adding and subtracting this value from the mean defines the confidence interval, which, in this case is $$12 \pm 1.2$$.

If we have a mean of $$\bar{x} = 12$$ for 10 samples with an experimental standard deviation of $$s = 2$$, then for the 95% confidence interval the value of $$t$$ and the resulting confidence interval are

t = qt(p = 0.025, 9) conf_int_samp = abs(t * 2/sqrt(10)) conf_int_samp

 1.430714

Adding and subtracting this value from the mean defines the confidence interval, which, in this case is $$12 \pm 1.4$$.

6.4: Using R to Find Confidence Intervals is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Harvey.