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Confidence Interval

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Confidence Interval

Confidence Interval

Confidence Interval

Table of Contents

Confidence Intervals: Complete Step-by-Step Tutorial on Point and Interval Estimation

1. Learning Objectives

By the end of this lecture, students should be able to:

  • Understand point estimation and interval estimation
  • Distinguish between a parameter, statistic, and estimator
  • Calculate point estimates for a population mean and proportion
  • Construct confidence intervals for a population mean
  • Decide when to use Z and t distributions
  • Construct a confidence interval for a population proportion
  • Interpret confidence intervals correctly
  • Calculate the margin of error
  • Determine the required sample size for estimating a mean or proportion
  • Understand the effect of sample size and confidence level on interval width

2. What Is Estimation?

In statistics, we usually want to learn something about a population, but collecting data from the entire population is often impossible.

For example, suppose a university has 10,000 students and we want to know their average monthly expenditure.

We cannot necessarily survey all 10,000 students.

Instead, we select a sample, for example:

n = 100

students.

We then use the sample information to estimate the population parameter.

The process is called statistical estimation.

There are two major types:

  1. Point estimation
  2. Interval estimation

3. Parameter vs. Statistic

This distinction is extremely important.

Population Parameter

A parameter describes the population.

Examples:

μ = population mean   |   σ = population standard deviation   |   p = population proportion

Usually, the parameter is unknown.

Sample Statistic

A statistic describes the sample.

Examples:

x̄ = sample mean   |   s = sample standard deviation   |   p̂ = sample proportion

We use these sample statistics to estimate unknown population parameters.

4. Point Estimation

Definition

A point estimate is a single numerical value used to estimate an unknown population parameter.

For example, suppose we want to estimate the average income of a population.

We take a sample and obtain:

x̄ = 65,000

Our point estimate of the population mean is:

μ ≈ 65,000

We are using one number to estimate the unknown population mean.

5. Common Point Estimators

Population Parameter Point Estimator
Population mean (μ)Sample mean (x̄)
Population proportion (p)Sample proportion (p̂)
Population variance (σ²)Sample variance (s²)
Population standard deviation (σ)Sample SD (s)

For today’s lecture, the most important are:

x̄ estimates μ   and   p̂ estimates p

6. Example 1: Point Estimate of a Population Mean

Suppose a researcher selects 8 students and records their weekly study hours:

8, 10, 7, 12, 9, 11, 6, 13

Step 1: Calculate the sample mean

x̄ = Σx / n

The total is:

8 + 10 + 7 + 12 + 9 + 11 + 6 + 13 = 76

Therefore:

x̄ = 76 / 8 = 9.5

Answer

The point estimate of the population’s average weekly study time is:

9.5 hours

Notice that we do not know the actual population mean.

We are using:

x̄ = 9.5

as our estimate.

7. Why Is a Point Estimate Not Enough?

Suppose Student A takes a sample of 100 students and obtains:

x̄ = 72

Student B takes another sample of 100 students and obtains:

x̄ = 75

Student C obtains:

x̄ = 70

Which value is the true population mean?

We do not know.

A point estimate provides one best estimate, but it does not tell us how much uncertainty surrounds that estimate.

This is why we need interval estimation.

8. Interval Estimation

Instead of giving one value, we provide a range of plausible values for the population parameter.

For example:

69.5 < μ < 74.5

This is an interval estimate.

A confidence interval normally has the form:

Point Estimate ± Margin of Error

Therefore:

CI = Estimate ± Margin of Error

This is the central idea behind confidence intervals.

9. What Is a Confidence Interval?

A confidence interval (CI) is a range of values calculated from sample data that is designed to capture an unknown population parameter with a specified level of confidence.

Common confidence levels are:

  • 90%
  • 95%
  • 99%

The most commonly used level is:

95%

10. The Basic Structure of a Confidence Interval

Every confidence interval can be understood as:

Estimate ± Critical Value × Standard Error

Therefore:

CI = Point Estimate ± Margin of Error

where:

ME = Critical Value × SE

So the calculation has three major components:

1. Point estimate

What is our best estimate?

2. Standard error

How much sampling variability exists?

3. Critical value

How much uncertainty do we want to cover?

11. Confidence Level and Critical Values

For the standard normal distribution:

Confidence LevelCritical Value (zα/2)
90%1.645
95%1.960
99%2.576

For example, at 95% confidence:

zα/2 = 1.96

12. Why Do We Use α?

If the confidence level is 95%:

Confidence = 0.95

The remaining probability is:

α = 1 − 0.95 = 0.05

Because the interval is two-sided, this 5% is divided between the two tails:

α/2 = 0.025

Therefore:

P(−1.96 < Z < 1.96) = 0.95

This gives the familiar 95% confidence interval.

13. Confidence Interval for a Population Mean

There are two important cases.

Case 1: Population standard deviation (σ) is known

Use the Z distribution:

x̄ ± zα/2 × (σ / √n)

Case 2: Population standard deviation (σ) is unknown

Use the t distribution:

x̄ ± tα/2, n−1 × (s / √n)

where:

  • x̄ = sample mean
  • σ = population standard deviation
  • s = sample standard deviation
  • n = sample size
  • zα/2 = Z critical value
  • tα/2, n−1 = t critical value
  • n − 1 = degrees of freedom

14. Example 2: 95% Confidence Interval When σ Is Known

Suppose:

x̄ = 72   |   σ = 10   |   n = 64

Construct a 95% confidence interval for the population mean.

Step 1: Identify the information

x̄ = 72   |   σ = 10   |   n = 64

Confidence level: 95%

Since σ is known, we use the Z distribution.

Step 2: Find the critical value

For 95% confidence:

zα/2 = 1.96

Step 3: Calculate the standard error

SE = σ / √n = 10 / √64 = 10 / 8 = 1.25

Step 4: Calculate the margin of error

ME = zα/2 × SE = 1.96 × 1.25 = 2.45

Step 5: Calculate the confidence interval

CI = x̄ ± ME = 72 ± 2.45

Lower limit: 72 − 2.45 = 69.55

Upper limit: 72 + 2.45 = 74.45

69.55 < μ < 74.45
Interpretation: We are 95% confident that the population mean lies between 69.55 and 74.45.

15. Example 3: Confidence Interval When σ Is Unknown

Suppose a researcher studies the monthly spending of 25 students.

The results are:

n = 25   |   x̄ = 68   |   s = 12

Construct a 95% confidence interval for the population mean.

Step 1: Determine whether to use Z or t

The population standard deviation is unknown.

We have: s = 12

Therefore, use the t distribution.

Step 2: Calculate degrees of freedom

df = n − 1 = 25 − 1 = 24

Step 3: Find the critical t-value

For 95% confidence and df = 24, the critical value is approximately:

t = 2.064

Step 4: Calculate the standard error

SE = s / √n = 12 / √25 = 12 / 5 = 2.4

Step 5: Calculate the margin of error

ME = t × SE = 2.064 × 2.4 = 4.9536 ≈ 4.95

Step 6: Calculate the confidence interval

CI = 68 ± 4.95

Lower limit: 68 − 4.95 = 63.05

Upper limit: 68 + 4.95 = 72.95

63.05 < μ < 72.95
Interpretation: We are 95% confident that the population mean lies between 63.05 and 72.95.

16. Z vs. t: What Should Students Remember?

Use this simple decision process:

Population SD known? → Use Z
Population SD unknown? → Use t

For the standard one-sample mean problem, when σ is unknown, the t-distribution is the appropriate choice.

The t-distribution has heavier tails than the standard normal distribution, reflecting the additional uncertainty from estimating σ using s.

17. Confidence Interval for a Population Proportion

Suppose the population parameter of interest is a proportion:

p

The sample proportion is:

p̂ = x / n

where:

  • x = number of successes
  • n = sample size

The approximate confidence interval is:

p̂ ± zα/2 × √(p̂(1 − p̂) / n)

18. Example 4: Confidence Interval for a Proportion

Suppose a university surveys 200 students.

Out of 200 students, 120 say that they prefer online learning.

Find the 95% confidence interval for the population proportion.

Step 1: Calculate the sample proportion

p̂ = x / n = 120 / 200 = 0.60

Therefore, 60% of the sample prefers online learning.

Step 2: Find the critical value

For 95% confidence:

z = 1.96

Step 3: Calculate the standard error

SEp̂ = √(p̂(1 − p̂) / n) = √((0.60 × 0.40) / 200) = √(0.24 / 200) = √0.0012 ≈ 0.03464

Step 4: Calculate the margin of error

ME = 1.96 × 0.03464 ≈ 0.06790

Step 5: Calculate the confidence interval

CI = 0.60 ± 0.06790

Lower limit: 0.60 − 0.06790 = 0.53210

Upper limit: 0.60 + 0.06790 = 0.66790

0.5321 < p < 0.6679

In percentage terms:

53.21% < p < 66.79%
Interpretation: We are 95% confident that the population proportion of students who prefer online learning is between 53.21% and 66.79%.

19. Complete Confidence Interval Formula Sheet

Population Mean — σ Known

x̄ ± zα/2 × (σ / √n)

Population Mean — σ Unknown

x̄ ± tα/2, n−1 × (s / √n)

Population Proportion

p̂ ± zα/2 × √(p̂(1 − p̂) / n)

20. Margin of Error

The margin of error (ME) tells us how far the confidence interval extends on either side of the point estimate.

Generally:

ME = Critical Value × SE

For a mean with known σ:

ME = zα/2 × (σ / √n)

For a mean with unknown σ:

ME = tα/2, n−1 × (s / √n)

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