Confidence Interval for a Mean

This lesson describes how to create a confidence interval for a population mean.

Checking Conditions

Before we can create a confidence interval for a population mean, we first need to make sure the following conditions are met to ensure that our confidence interval will be accurate:

  • Random: A random sample or random experiment should be used to collect the data.
  • Normal: The sampling distribution of the sample mean needs to be approximately normal. This is true if the population we are studying is normal or if our sample size is sufficiently large (typically n ≥ 30)
  • Independent: Individual observations are independent. Typically this holds true if a randomized method is used to collect the data.

If these three conditions are met, then we can construct a confidence interval for a population mean using the distribution.

The formula to find a confidence interval for a population mean is:

x +/- tn-1 * (s / √n)

where x is the sample mean, tn-1 is the t critical-value that comes from the t distribution table with n-1 degrees of freedom, s is the sample standard deviation, and n is the sample size. 

Let’s walk through some examples of how to find a confidence interval for a population mean.

Examples of Finding a Confidence Interval for a Population Mean

Example 1: Bob wants to know the mean height of tomato plants in a certain region with 1,000 individual plants. He collects a random sample of 25 plants. The sample data were roughly symmetric with a mean of 8 inches and a standard deviation of 2 inches. Find a 95% confidence interval for the mean height of tomato plants in this region.

Step 1: Identify the sample mean, sample standard deviation, and sample size.

Sample mean (x ) = 8 inches
Sample standard deviation (s) = 2 inches
Sample size (n) = 25 tomato plants

Step 2: Identify the t-critical value

To find the t-critical value (tn-1) we need to identify the degrees of freedom and the confidence level.

The degrees of freedom = n-1 = 25-1 = 24

The confidence level = 0.95

Using the t-table, we look up 1-0.95 = 0.05 in the two-tails row that corresponds to 24 degrees of freedom:

The t-critical value is 2.064.

Step 3: Plug these numbers into the formula to find the confidence interval for a population mean.

Confidence interval:  x +/- tn-1 * (s / √n)

Confidence interval: 8 +/ (2.064) * (2 / √25)

Confidence interval: 8 +/- (.8256) = (7.1744 , 8.8256)

The 95% confidence interval for the mean height (in inches) of tomato plants in this region is (7.1744 , 8.8256).

Example 2: Sarah wants to know the mean weight of a burger at a certain restaurant that produces 500 burgers per day. She collects a random sample of 15 burgers and records their weights. The sample data were roughly symmetric with a mean of .23 ounces and a standard deviation of 0.3 ounces. Find a 99% confidence interval for the mean weight of a burger at this restaurant.

Step 1: Identify the sample mean, sample standard deviation, and sample size.

Sample mean (x ) = .23 ounces
Sample standard deviation (s) = .03 ounces
Sample size (n) = 15 burgers

Step 2: Identify the t-critical value

To find the t-critical value (tn-1) we need to identify the degrees of freedom and the confidence level.

The degrees of freedom = n-1 = 15-1 = 14

The confidence level = 0.99

Using the t-table, we look up 1-0.99 = 0.01 in the two-tails row that corresponds to 14 degrees of freedom:

The t-critical value is 2.145.

Step 3: Plug these numbers into the formula to find the confidence interval for a population mean.

Confidence interval:  x +/- tn-1 * (s / √n)

Confidence interval: .23 +/ (2.145) * (.03 / √15)

Confidence interval: .23 +/- (.0166) = (.2134 , .2466)

The 99% confidence interval for the mean weight (in ounces) of a burger from this restaurant is (.2134 , .2466).

How to Interpret a Confidence Interval for a Population Mean

After we build a confidence interval for a population mean, it’s important to be able to interpret what the interval actually tells us.

A confidence interval for a mean gives us a range of plausible value for the true population mean. If a confidence interval doesn’t contain a particular value, we can say that it’s unlikely that the particular value is equal to the true population mean.

Knowing this, let’s walk through a few examples of how to interpret a confidence interval for a population mean.

Example 1: In a previous example, Bob found the 95% confidence interval for the mean height (in inches) of tomato plants in a certain region to be (7.1744 , 8.8256). Based on this interval, is it plausible that the mean height of all tomato plants in this region is 9 inches?

No, it isn’t. The interval tells us that the plausible values for the true mean height of tomato plants in this region is between 7.1744 and 8.8256 inches. Since 9 inches is not within this interval, it doesn’t seem plausible that the true mean height of tomato plants in this region could be 9 inches. 

Example 2: A particular restaurant advertises that their burgers have a mean weight of .24 ounces. In a previous example, Sarah found the 99% confidence interval for the mean weight (in ounces) of a burger at this restaurant to be (.2134 , .2466). Based on this interval, is it plausible that the mean weight of a burger at this this restaurant actually is .24 ounces?

Yes. Since 0.24 is within the 99% confidence interval, it is a plausible  value for the mean weight of a burger at this restaurant.

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