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A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence.

It is calculated using the following general formula:

Confidence Interval = (point estimate)  +/-  (critical value)*(standard error)

This formula creates an interval with a lower bound and an upper bound, which likely contains a population parameter with a certain level of confidence:

Confidence Interval  = [lower bound, upper bound]

This tutorial explains how to calculate the following confidence intervals in R:

1. Confidence Interval for a Mean

2. Confidence Interval for a Difference in Means

3. Confidence Interval for a Proportion

4. Confidence Interval for a Difference in Proportions

Let’s jump in!

Example 1: Confidence Interval for a Mean

We use the following formula to calculate a confidence interval for a mean:

Confidence Interval = x  +/-  t_{n-1, 1-α/2}*(s/√n)

where:

• x: sample mean
• t: the t-critical value
• s: sample standard deviation
• n: sample size

Example: Suppose we collect a random sample of turtles with the following information:

• Sample size n = 25
• Sample mean weight x = 300
• Sample standard deviation s = 18.5

The following code shows how to calculate a 95% confidence interval for the true population mean weight of turtles:

#input sample size, sample mean, and sample standard deviation
n <- 25
xbar <- 300 
s <- 18.5

#calculate margin of error
margin <- qt(0.975,df=n-1)*s/sqrt(n)

#calculate lower and upper bounds of confidence interval
low <- xbar - margin
low

[1] 292.3636

high <- xbar + margin
high

[1] 307.6364

The 95% confidence interval for the true population mean weight of turtles is [292.36, 307.64].

Example 2: Confidence Interval for a Difference in Means

We use the following formula to calculate a confidence interval for a difference in population means:

Confidence interval = (x₁–x₂) +/- t*√((s_p²/n₁) + (s_p²/n₂))

where:

• x₁, x₂: sample 1 mean, sample 2 mean
• t: the t-critical value based on the confidence level and (n₁+n₂-2) degrees of freedom
• s_p²: pooled variance, calculated as ((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)
• t: the t-critical value
• n₁, n₂: sample 1 size, sample 2 size

Example: Suppose we want to estimate the difference in mean weight between two different species of turtles, so we go out and gather a random sample of 15 turtles from each population. Here is the summary data for each sample:

Sample 1:

• x₁ = 310
• s₁ = 18.5
• n₁ = 15

Sample 2:

• x₂ = 300
• s₂ = 16.4
• n₂ = 15

The following code shows how to calculate a 95% confidence interval for the true difference in population means:

#input sample size, sample mean, and sample standard deviation
n1 <- 15
xbar1 <- 310 
s1 <- 18.5

n2 <- 15
xbar2 <- 300
s2 <- 16.4

#calculate pooled variance
sp = ((n1-1)*s1^2 + (n2-1)*s2^2) / (n1+n2-2)

#calculate margin of error
margin <- qt(0.975,df=n1+n2-1)*sqrt(sp/n1 + sp/n2)

#calculate lower and upper bounds of confidence interval
low <- (xbar1-xbar2) - margin
low

[1] -3.055445

high <- (xbar1-xbar2) + margin
high

[1] 23.05544

The 95% confidence interval for the true difference in population means is [-3.06, 23.06].

Example 3: Confidence Interval for a Proportion

We use the following formula to calculate a confidence interval for a proportion:

Confidence Interval = p  +/-  z*(√p(1-p) / n)

where:

• p: sample proportion
• z: the chosen z-value
• n: sample size

Example: Suppose we want to estimate the proportion of residents in a county that are in favor of a certain law. We select a random sample of 100 residents and ask them about their stance on the law. Here are the results:

• Sample size n = 100
• Proportion in favor of law p = 0.56

The following code shows how to calculate a 95% confidence interval for the true proportion of residents in the entire county who are in favor of the law:

#input sample size and sample proportion
n <- 100
p <- .56

#calculate margin of error
margin <- qnorm(0.975)*sqrt(p*(1-p)/n)

#calculate lower and upper bounds of confidence interval
low <- p - margin
low

[1] 0.4627099

high <- p + margin
high

[1] 0.6572901

The 95% confidence interval for the true proportion of residents in the entire county who are in favor of the law is [.463, .657].

Example 4: Confidence Interval for a Difference in Proportions

We use the following formula to calculate a confidence interval for a difference in proportions:

Confidence interval = (p₁–p₂)  +/-  z*√(p₁(1-p₁)/n₁ + p₂(1-p₂)/n₂)

where:

• p₁, p₂: sample 1 proportion, sample 2 proportion
• z: the z-critical value based on the confidence level
• n₁, n₂: sample 1 size, sample 2 size

Example: Suppose we want to estimate the difference in the proportion of residents who support a certain law in county A compared to the proportion who support the law in county B. Here is the summary data for each sample:

Sample 1:

• n₁ = 100
• p₁ = 0.62 (i.e. 62 out of 100 residents support the law)

Sample 2:

• n₂ = 100
• p₂ = 0.46 (i.e. 46 our of 100 residents support the law)

The following code shows how to calculate a 95% confidence interval for the true difference in proportion of residents who support the law between the counties:

#input sample sizes and sample proportions
n1 <- 100
p1 <- .62

n2 <- 100
p2 <- .46

#calculate margin of error
margin <- qnorm(0.975)*sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2)

#calculate lower and upper bounds of confidence interval
low <- (p1-p2) - margin
low

[1] 0.02364509


high <- (p1-p2) + margin
high

[1] 0.2963549

The 95% confidence interval for the true difference in proportion of residents who support the law between the counties is [.024, .296].

You can find more R tutorials here.