Confidence Interval for Mean
Learn how to estimate an unknown population mean using a sample mean, standard error, and an appropriate critical value.
What Are We Estimating?
In many analytics problems, the quantity we want to know is the average value for an entire population.
For example, a company may want to estimate the average spending of all its customers. Measuring every customer may not be practical, so we collect a sample.
Population
Unknown mean μ
Sample
Sample mean x̄
Confidence Interval
Range for μ
What You Will Learn
Why we construct a confidence interval for a mean
The role of the sample mean
Standard error of the mean
Critical values
The z-based confidence interval
The t-based confidence interval
How sample size affects the interval
How to interpret the result correctly
Start With the Sample Mean
Suppose an online store selects 100 customers and calculates their average monthly spending.
Sample size
n = 100
Sample mean
x̄ = ₹2,500
The sample mean of ₹2,500 is our point estimate of the unknown population mean μ.
Calculate the Standard Error
The standard error measures the typical sampling variability of the sample mean.
SE = σ / √n
When the population standard deviation σ is known.
SE = s / √n
In practice, when σ is unknown, the sample standard deviation s is commonly used.
Z-Based Confidence Interval
When the population standard deviation is known and the relevant assumptions are satisfied, a z-based interval can be written as:
x̄ ± z* × (σ / √n)
x̄
Sample mean
z*
Critical value
σ / √n
Standard error
A Worked Example
Suppose an analyst wants a 95% confidence interval for average customer spending.
Sample mean
₹2,500
σ
₹600
n
100
95% z*
1.96
1. Calculate Standard Error
SE = 600 / √100 = 600 / 10 = ₹60
2. Calculate Margin of Error
ME = 1.96 × 60 = ₹117.60
3. Construct the Interval
₹2,500 ± ₹117.60
₹2,382.40 to ₹2,617.60
What If σ Is Unknown?
In many real-world situations, the population standard deviation σ is unknown. Instead, we estimate it using the sample standard deviation s.
x̄ ± t* × (s / √n)
The t-distribution is used to obtain the critical value, with degrees of freedom typically equal to n − 1 for the one-sample mean procedure.
Practical rule:
When σ is unknown, the one-sample t procedure is commonly used, subject to its assumptions.
Why Does Sample Size Matter?
For a mean, the standard error contains √n in the denominator. Therefore, increasing the sample size generally decreases the standard error.
n = 25
SE = ₹120
Example when σ = ₹600.
n = 100
SE = ₹60
Four times the sample size halves the SE.
n = 400
SE = ₹30
More data generally gives greater precision.
Z vs t: The Basic Difference
The choice between procedures depends on what is known and on the assumptions of the problem. In typical one-sample mean inference, the t procedure is especially important when σ is unknown.
Interpreting the Result
Example:
95% CI = ₹2,382.40 to ₹2,617.60
The confidence interval procedure is designed so that, over many repeated random samples under the same conditions, about 95% of intervals constructed this way would contain the true population mean.
Avoid This Interpretation
Do not describe a 95% confidence interval as meaning that there is a 95% probability that the fixed population mean is inside this particular interval in the frequentist framework.
Delivery Time Analysis
A food delivery company wants to estimate the average delivery time for all orders in a city.
Sample size
100
Mean
32 min
Standard deviation
8 min
Confidence
95%
Using a t-based procedure:
SE = 8 / √100 = 0.8 minutes
The appropriate t critical value depends on the confidence level and degrees of freedom. The final interval would be centered at the sample mean of 32 minutes.
Test Your Understanding
CHECK YOUR UNDERSTANDING
What population quantity is being estimated in a confidence interval for a mean?
CHECK YOUR UNDERSTANDING
Which value is the point estimate of the population mean?
CHECK YOUR UNDERSTANDING
If σ = 600 and n = 100, what is the standard error of the sample mean?
CHECK YOUR UNDERSTANDING
When the population standard deviation is unknown, which procedure is commonly used for a one-sample mean?
CHECK YOUR UNDERSTANDING
What generally happens to standard error when sample size increases?
CHECK YOUR UNDERSTANDING
A sample mean is ₹2,500 and the margin of error is ₹100. What is the interval?
Fill in the Blanks
The sample ______ is commonly used as the point estimate of the population mean.
For a known population standard deviation, the standard error of the mean is σ divided by the square root of ______.
When σ is unknown, the one-sample ______ procedure is commonly used.
A confidence interval is centered around the sample ______.
Website Response Time
A sample of 100 website requests has a mean response time of 250 milliseconds. Assume the population standard deviation is known to be 40 milliseconds.
Calculate the standard error.
SE = 40 / √100
SE = 4 milliseconds
Using z* = 1.96 for a 95% interval, calculate the margin of error.
ME = 1.96 × 4 = 7.84 milliseconds
95% CI = 242.16 ms to 257.84 ms
Final Challenge
Two studies estimate the same population mean using the same confidence level.
Study A
n = 100
Margin of error = ₹120
Study B
n = 400
Margin of error = ₹60
What does this demonstrate?
With the other relevant factors held constant, the larger sample can produce a smaller standard error and therefore a narrower confidence interval.
Lesson Summary
A confidence interval for a mean estimates the unknown population mean μ.
The sample mean x̄ is the point estimate.
Standard error measures sampling variability of the sample mean.
When σ is known, a z-based procedure can be used under the appropriate assumptions.
When σ is unknown, the one-sample t procedure is commonly used.
Larger samples generally reduce standard error and improve precision.
The confidence interval is centered on the sample mean.
Correct interpretation of a confidence interval refers to the long-run performance of the interval-producing procedure.