Introduction
Confidence intervals are a fundamental concept in inferential statistics. They provide a range of values that likely contain a population parameter, such as a mean or proportion, with a certain level of confidence. This topic will cover the following:
- Definition and interpretation of confidence intervals
- Calculating confidence intervals for means and proportions
- Practical examples and exercises
- Definition and Interpretation of Confidence Intervals
What is a Confidence Interval?
A confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval.
Key Concepts
- Point Estimate: A single value estimate of a population parameter (e.g., sample mean).
- Margin of Error (MoE): The range above and below the point estimate.
- Confidence Level: The probability that the interval contains the population parameter (commonly 90%, 95%, or 99%).
Interpretation
A 95% confidence interval means that if we were to take 100 different samples and compute a CI for each sample, we would expect about 95 of the intervals to contain the population parameter.
- Calculating Confidence Intervals
Confidence Interval for a Population Mean
When the population standard deviation (\(\sigma\)) is known, the confidence interval for the population mean (\(\mu\)) is calculated as follows:
\[ \text{CI} = \bar{x} \pm Z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \(\bar{x}\) = sample mean
- \(Z_{\alpha/2}\) = Z-value from the standard normal distribution for the desired confidence level
- \(\sigma\) = population standard deviation
- \(n\) = sample size
Example
Suppose we have a sample mean (\(\bar{x}\)) of 50, a population standard deviation (\(\sigma\)) of 10, and a sample size (\(n\)) of 30. We want to calculate a 95% confidence interval.
- Find the Z-value for a 95% confidence level (Z = 1.96).
- Calculate the margin of error (MoE):
\[ \text{MoE} = 1.96 \left(\frac{10}{\sqrt{30}}\right) \approx 3.58 \]
- Calculate the confidence interval:
\[ \text{CI} = 50 \pm 3.58 \] \[ \text{CI} = (46.42, 53.58) \]
Confidence Interval for a Population Proportion
The confidence interval for a population proportion (\(p\)) is calculated as follows:
\[ \text{CI} = \hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
Where:
- \(\hat{p}\) = sample proportion
- \(Z_{\alpha/2}\) = Z-value from the standard normal distribution for the desired confidence level
- \(n\) = sample size
Example
Suppose we have a sample proportion (\(\hat{p}\)) of 0.6, and a sample size (\(n\)) of 100. We want to calculate a 95% confidence interval.
- Find the Z-value for a 95% confidence level (Z = 1.96).
- Calculate the margin of error (MoE):
\[ \text{MoE} = 1.96 \sqrt{\frac{0.6(1 - 0.6)}{100}} \approx 0.096 \]
- Calculate the confidence interval:
\[ \text{CI} = 0.6 \pm 0.096 \] \[ \text{CI} = (0.504, 0.696) \]
- Practical Examples and Exercises
Example 1: Confidence Interval for a Mean
A researcher wants to estimate the average height of adult males in a city. A random sample of 50 adult males has an average height of 175 cm with a standard deviation of 8 cm. Calculate the 95% confidence interval for the population mean height.
Solution:
- Sample mean (\(\bar{x}\)) = 175 cm
- Sample standard deviation (\(s\)) = 8 cm
- Sample size (\(n\)) = 50
- Z-value for 95% confidence level = 1.96
\[ \text{MoE} = 1.96 \left(\frac{8}{\sqrt{50}}\right) \approx 2.21 \]
\[ \text{CI} = 175 \pm 2.21 \] \[ \text{CI} = (172.79, 177.21) \]
Example 2: Confidence Interval for a Proportion
A survey finds that 45 out of 100 people prefer a new product over the old one. Calculate the 90% confidence interval for the population proportion.
Solution:
- Sample proportion (\(\hat{p}\)) = 0.45
- Sample size (\(n\)) = 100
- Z-value for 90% confidence level = 1.645
\[ \text{MoE} = 1.645 \sqrt{\frac{0.45(1 - 0.45)}{100}} \approx 0.081 \]
\[ \text{CI} = 0.45 \pm 0.081 \] \[ \text{CI} = (0.369, 0.531) \]
Exercises
Exercise 1
A sample of 40 students has an average test score of 78 with a standard deviation of 10. Calculate the 99% confidence interval for the population mean test score.
Exercise 2
In a poll, 120 out of 200 respondents said they support a new policy. Calculate the 95% confidence interval for the population proportion.
Solutions
Solution to Exercise 1
- Sample mean (\(\bar{x}\)) = 78
- Sample standard deviation (\(s\)) = 10
- Sample size (\(n\)) = 40
- Z-value for 99% confidence level = 2.576
\[ \text{MoE} = 2.576 \left(\frac{10}{\sqrt{40}}\right) \approx 4.07 \]
\[ \text{CI} = 78 \pm 4.07 \] \[ \text{CI} = (73.93, 82.07) \]
Solution to Exercise 2
- Sample proportion (\(\hat{p}\)) = 0.6
- Sample size (\(n\)) = 200
- Z-value for 95% confidence level = 1.96
\[ \text{MoE} = 1.96 \sqrt{\frac{0.6(1 - 0.6)}{200}} \approx 0.068 \]
\[ \text{CI} = 0.6 \pm 0.068 \] \[ \text{CI} = (0.532, 0.668) \]
Conclusion
In this section, we have covered the concept of confidence intervals, their interpretation, and how to calculate them for both population means and proportions. Confidence intervals are a powerful tool in statistics, providing a range of values that likely contain the population parameter. Understanding and correctly interpreting confidence intervals is crucial for making informed decisions based on sample data.
Next, we will delve into the topic of Correlation Analysis in Module 6, where we will explore the relationships between different variables.