How to Calculate Confidence Intervals: 3 Example Problems

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

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

Confidence Interval = x +/- t*(s/√n)

where:

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

Note: We replace a t critical value with a z critical value in the formula if the population standard deviation (σ) is known and the sample size is greater than 30.

The following examples show how to construct a confidence interval for a mean in three different scenarios:

Population standard deviation (σ) is unknown
Population standard deviation (σ) is known but n ≤ 30
Population standard deviation (σ) is known and n > 30

Let’s jump in!

Example 1: Confidence Interval when σ is Unknown

Suppose we would like to calculate a 95% confidence interval for the mean height (in inches) of a certain species of plant.

Suppose we collect a simple random sample with the following information:

sample mean (x)= 12
sample size (n) = 19
sample standard deviation (s) = 6.3

We can use the following formula to construct this confidence interval:

95% C.I. = x +/- t*(s/√n)
95% C.I. = 12 +/- t_{n-1, α/2}*(6.3/√19)
95% C.I. = 12 +/- t_{18, .025}*(6.3/√19)
95% C.I. = 12 +/- 2.1009*(6.3/√19)
95% C.I. = (8.964 , 15.037)

The 95% confidence interval for the population mean height for this particular species of plant is (8.964 inches, 15.037 inches).

Note #1: We used the Inverse t Distribution Calculator to find the t critical value associated with 18 degrees of freedom and a confidence level of 0.95.

Note #2: Since the population standard deviation (σ) was unknown, we used the t critical value when calculating the confidence interval.

Example 2: Confidence Interval when σ is Known but n ≤ 30

Suppose we would like to calculate a 99% confidence interval for the mean exam score on a certain college entrance exam.

Suppose we collect a simple random sample with the following information:

sample mean (x)= 85
sample size (n) = 25
population standard deviation (σ) = 3.5

We can use the following formula to construct this confidence interval:

99% C.I. = x +/- t*(s/√n)
99% C.I. = 85 +/- t_{n-1, α/2}*(3.5/√25)
99% C.I. = 85 +/- t_{24, .005}*(3.5/√25)
99% C.I. = 85 +/- 2.7969*(3.5/√25)
99% C.I. = (83.042, 86.958)

The 99% confidence interval for the population mean exam score on this particular college entrance exam is (83.042, 86.958).

Note #1: We used the Inverse t Distribution Calculator to find the t critical value associated with 24 degrees of freedom and a confidence level of 0.99.

Note #2: Since the population standard deviation (σ) was known but the sample size (n) was less than 30, we used the t critical value when calculating the confidence interval.

Example 3: Confidence Interval when σ is Known and n > 30

Suppose we would like to calculate a 90% confidence interval for the mean weight of a certain species of turtle.

Suppose we collect a simple random sample with the following information:

sample mean (x)= 300
sample size (n) = 40
population standard deviation (σ) = 15

We can use the following formula to construct this confidence interval:

90% C.I. = x +/- z*(σ/√n)
90% C.I. = 300 +/- 1.645*(15/√40)
90% C.I. = (296.099, 303.901)

The 90% confidence interval for the population mean weight of this particular species of turtle is (83.042, 86.958).

Note #1: We used the Critical Z Value Calculator to find the z critical value associated with a significance level of 0.1.

Note #2: Since the population standard deviation (σ) was known and the sample size (n) was greater than 30, we used the z critical value when calculating the confidence interval.