Lecture 8(1)
Chapter 8: Interval Estimation
8.1: Population Mean, σ Known
Introduction to Interval Estimation
Point estimates usually do not provide exact values of population parameters.
Interval estimators provide an interval where true parameters likely lie, with a specified degree of confidence.
Formula: Interval estimate = Point estimate ± Margin of error.
Interval Estimation of Population Mean:
Form:
Interval estimate of mean: ( \bar{x} \pm \text{margin of error} ).
Margin of error is calculated using:
Population standard deviation (σ)
Sample standard deviation (s).
Note: σ is often not known but can be estimated.
Confidence Levels and Margins of Error
1 − α is the probability that a sample mean will provide a specific margin of error of z({\alpha/2})σ({\bar{x}}).
Fixed confidence level determines α, 1 − α, z(_{\alpha/2}), influencing the margin of error.
Formula for interval estimate:
( \bar{x} \pm z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right) )
8.2: Population Mean, σ Unknown
When σ is unknown, we use sample standard deviation (s).
This introduces the application of Student's t-distribution, which is relevant for smaller sample sizes.
The degrees of freedom (df) for t-distribution equals n − 1.
Larger n leads to t-distribution approximating normal distribution.
Interval Estimation Formula when σ is Unknown:
( \bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right) )
8.3: Determining the Sample Size
Margin of Error (E):
E is defined as the upper and lower bounds of the interval estimate.
Formulae:
( E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}} )
Necessary sample size:
( n = \left(\frac{z_{\alpha/2} \sigma}{E}\right)^{2} )
8.4: Population Proportion
Interval Estimate of Population Proportion:
General form: ( \bar{p} \pm \text{margin of error} ).
Sampling distribution approximated as normal given conditions np ≥ 5 and n(1 − p) ≥ 5 are met.
Interval Estimate Formula for Proportion:
( \bar{p} \pm z_{\alpha/2} \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}} )
8.5: Examples and Problem Solving
Example: Discount Sounds
Sample size n = 36, mean income = $41,100, σ = $4,500.
At 95% confidence, the margin of error is calculated leading to the interval estimate of mean income.
Example: Apartment Rents
Sample of 16 one-bedroom apartments, mean rent = $750, s = $55, 95% confidence.
Derived confidence interval using t-distribution due to unknown σ.
Chapter Recap
Key takeaway: Interval estimation allows assessment of how close sample estimates (mean and proportion) are to their true population parameters using predetermined levels of confidence based on the Central Limit Theorem.
Ensure random sampling to avoid bias in estimates and confidence intervals.
Lecture 8(1)
Chapter 8: Interval Estimation
8.1: Population Mean, σ Known
Introduction to Interval Estimation
Point estimates usually do not provide exact values of population parameters.
Interval estimators provide an interval where true parameters likely lie, with a specified degree of confidence.
Formula: Interval estimate = Point estimate ± Margin of error.
Interval Estimation of Population Mean:
Form:
Interval estimate of mean: ( \bar{x} \pm \text{margin of error} ).
Margin of error is calculated using:
Population standard deviation (σ)
Sample standard deviation (s).
Note: σ is often not known but can be estimated.
Confidence Levels and Margins of Error
1 − α is the probability that a sample mean will provide a specific margin of error of z({\alpha/2})σ({\bar{x}}).
Fixed confidence level determines α, 1 − α, z(_{\alpha/2}), influencing the margin of error.
Formula for interval estimate:
( \bar{x} \pm z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right) )
8.2: Population Mean, σ Unknown
When σ is unknown, we use sample standard deviation (s).
This introduces the application of Student's t-distribution, which is relevant for smaller sample sizes.
The degrees of freedom (df) for t-distribution equals n − 1.
Larger n leads to t-distribution approximating normal distribution.
Interval Estimation Formula when σ is Unknown:
( \bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right) )
8.3: Determining the Sample Size
Margin of Error (E):
E is defined as the upper and lower bounds of the interval estimate.
Formulae:
( E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}} )
Necessary sample size:
( n = \left(\frac{z_{\alpha/2} \sigma}{E}\right)^{2} )
8.4: Population Proportion
Interval Estimate of Population Proportion:
General form: ( \bar{p} \pm \text{margin of error} ).
Sampling distribution approximated as normal given conditions np ≥ 5 and n(1 − p) ≥ 5 are met.
Interval Estimate Formula for Proportion:
( \bar{p} \pm z_{\alpha/2} \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}} )
8.5: Examples and Problem Solving
Example: Discount Sounds
Sample size n = 36, mean income = $41,100, σ = $4,500.
At 95% confidence, the margin of error is calculated leading to the interval estimate of mean income.
Example: Apartment Rents
Sample of 16 one-bedroom apartments, mean rent = $750, s = $55, 95% confidence.
Derived confidence interval using t-distribution due to unknown σ.
Chapter Recap
Key takeaway: Interval estimation allows assessment of how close sample estimates (mean and proportion) are to their true population parameters using predetermined levels of confidence based on the Central Limit Theorem.
Ensure random sampling to avoid bias in estimates and confidence intervals.
