Confidence Interval Estimation
Concepts in Interval Estimation
- Inferential Statistics Objective: To estimate population characteristics using sample information.
- Point Estimate: A sample characteristic (e.g., sample mean) used to estimate the corresponding population characteristic.
- Margin of Error: A range of values around the point estimate, within which the population value is estimated to fall.
- Confidence Level: The degree of certainty associated with the estimate.
- Interval Estimate: A range of values within which the population value is believed to fall. Also known as the confidence interval.
- Example Scenario:
- Estimating the average amount spent on lunch by UGH students (population mean).
- Surveying 200 students (sample size).
- Sample reveals an average lunch spending of $12 (sample mean).
- Inferring that all students spend an average of $12 (point estimate).
- Applying a margin of error of +/- $3.
- Deriving a range from $9 to $15 (interval estimate).
- Expressing 95% confidence in the estimation (confidence level).
Key Terms
- Point Estimate: A single value derived from the sample to estimate the population value (typically the sample mean).
- Margin of Error: Measures the maximum expected difference between sample results and the actual population values.
- Interval Estimate: A range of values used to estimate the population characteristic, determined by adding and subtracting the margin of error from the point estimate (confidence interval).
- Confidence Level: The probability that the interval estimate contains the actual population mean (parameter).
Methods in Interval Estimation
- Z-Distribution: Used when the population standard deviation is known, OR when the sample size is at least 30.
- T-Distribution: Used when the population standard deviation is unknown AND the sample size is less than 30.
- Formulas:
- Mean \pm (z) (\frac{SD}{\sqrt{n}})
- Mean \pm (t) (\frac{SD}{\sqrt{n}})
Critical Values for Z-Distribution
| Confidence Level | Alpha Two Tail | Alpha One Tail | Critical Value |
|---|
| 80% | 20% | 10% | 1.28 |
| 90% | 10% | 5% | 1.65 |
| 95% | 5% | 2.5% | 1.96 |
| 98% | 2% | 1% | 2.33 |
| 99% | 1% | 0.05% | 2.58 |
Critical Values for T-Distribution
| Confidence Level | Alpha Two-Tail | Alpha One-Tail | Deg. of Freedom (n - 1) | Critical Value |
|---|
| 80% | 20% | 10% | 1 | 3.078 |
| 90% | 10% | 5% | 1 | 6.314 |
| 95% | 5% | 2.5% | 1 | 12.706 |
| 98% | 2% | 1% | 1 | 31.821 |
| 99% | 1% | 0.5% | 1 | 63.657 |
| 80% | 20% | 10% | 2 | 1.886 |
| 90% | 10% | 5% | 2 | 2.920 |
| 95% | 5% | 2.5% | 2 | 4.303 |
| 98% | 2% | 1% | 2 | 6.965 |
| 99% | 1% | 0.5% | 2 | 9.925 |
| 80% | 20% | 10% | 3 | 1.638 |
| 90% | 10% | 5% | 3 | 2.353 |
| 95% | 5% | 2.5% | 3 | 3.182 |
| 98% | 2% | 1% | 3 | 4.541 |
| 99% | 1% | 0.5% | 3 | 5.841 |
| 80% | 20% | 10% | 4 | 1.533 |
| 90% | 10% | 5% | 4 | 2.132 |
| 95% | 5% | 2.5% | 4 | 2.776 |
| 98% | 2% | 1% | 4 | 3.747 |
| 99% | 1% | 0.5% | 4 | 4.604 |
The table continues for degrees of freedom 5 through 10.
Interval Estimate for Z-Distribution - Example 1
- Problem: Estimating the average hourly fee for counseling services in the city.
- Given:
- Sample Mean = $60
- Standard Deviation = $12
- Sample Size = 50
- Confidence Level = 95% (Z value = 1.96)
- Solution:
- (a) Point Estimate: $60 (sample mean)
- (b) Margin of Error:
- (\frac{(1.96)(12)}{\sqrt{50})} = $3.33
- (c) Interval Estimate:
- Lower = 60 – 3.33 = $56.67
- Upper = 60 + 3.33 = $63.33
- (d) Interpretation: We are 95% confident that the true average hourly wage for all counselors in the city lies between $56.67 and $63.33.
Interval Estimate for Z-Distribution - Example 2
- Problem: Estimating the average time it takes the police to respond to emergency calls.
- Given:
- Sample Mean = 450 seconds
- Standard Deviation = 60 seconds
- Sample Size = 120
- Confidence Level = 90% (Z value = 1.65)
- Solution:
- (a) Point Estimate: 450 seconds
- (b) Margin of Error:
- (\frac{(1.65)(60)}{\sqrt{120})} = 9.04
- (c) Interval Estimate:
- Lower = 450 – 9.04 = 440.96 seconds
- Upper = 450 + 9.04 = 459.04 seconds
- (d) Interpretation: We are 90% confident that the true average response time for all emergency calls lies between 440.96 and 459.04 seconds.
The Effect of Sample Size
- Increasing the sample size from 120 to 480 (4 times increase):
- Margin of Error decreases.
- Example:
- Sample Size = 120, Margin of Error = 9.04, Interval Estimate = 440.96 and 459.04
- Sample Size = 480, Margin of Error = 4.52, Interval Estimate = 445.48 and 454.52
- When the sample size increases four times its original size, the margin of error will reduce by half.
- Conversely, when the sample size is reduced by a quarter of its original size, then the margin of error will become twice as large.
The Effect of Confidence Level
- Increasing the confidence level:
- The z-value becomes larger.
- The margin of error becomes larger.
- The interval estimate becomes wider.
- Estimate becomes less precise.
- Z-values for different confidence levels:
- 90% (z-value = \pm 1.65)
- 95% (z-value = \pm 1.96)
- 99% (z-value = \pm 2.58)
- Need a delicate balance between confidence and precision.
- All being equal:
- An increase in confidence level or an increase in standard deviation will increase the margin of error.
- However, an increase in sample size will decrease the margin of error.
Interval Estimate for Z-Distribution - Example 3 (Effect of Confidence Level)
- Problem: What effect does increasing the confidence level from 90% to 99% have on the margin of error?
- Given:
- Sample Mean = 450
- Standard Deviation = 60
- Sample Size = 480
- Confidence Level = 99% (Z value = 2.58)
- Solution:
- (a) Point Estimate: 450
- (b) Margin of Error:
- (\frac{(2.58)(60)}{\sqrt{480})} = 7.07
- (c) Interval Estimate:
- Lower = 450 – 7.07 = 442.93
- Upper = 450 + 7.07 = 457.07
- (d) Interpretation: We are 99% confident that the true average response time for all emergency calls will lie between 442.93 and 457.07 seconds.
Interval Estimate for T-Distribution
- When the population standard deviation is unknown and the sample size is less than 30, the T-distribution is used.
- The t-distribution is influenced by the degree of freedom (n – 1).
- T-critical value is obtained from t-distribution table under various degrees of freedom and confidence levels.
- The t-distribution resembles the normal distribution but is more spread out, and the height of the curve is lower.
- As the sample size increases, the t-distribution approaches the standard normal z-distribution. The approximation is quite close when the sample size is 30 or more.
Interval Estimate for T-Distribution - Example 1
- Problem: Estimating the average salary of social workers in a town.
- Given:
- Sample Mean = $45,500
- Standard Deviation = $12,500
- Sample Size = 20
- Confidence Level = 90%
- Degrees of Freedom = 20 – 1 = 19 (T value = 1.729)
- Solution:
- (a) Point Estimate: $45,500
- (b) Margin of Error:
- (\frac{(1.729)(12500)}{\sqrt{20})} = 4832.70
- (c) Interval Estimate:
- Lower = 45500 – 4832.70 = $40,667.30
- Upper = 45500 + 4832.70 = $50,332.70
- (d) Interpretation: We are 90% confident that the true average salary for all social workers in the entire town will lie between $40,667.30 and $50,332.70.
Interval Estimate for T-Distribution - Example 2
- Problem: Estimating the average amount spent on food by households in a community.
- Given:
- Sample Mean = $520
- Standard Deviation = $45
- Sample Size = 24
- Confidence Level = 95%
- Degrees of Freedom = 24 – 1 = 23 (T value = 2.069)
- Solution:
- (a) Point Estimate: $520
- (b) Margin of Error:
- (\frac{(2.069)(45)}{\sqrt{24})} = $19
- (c) Interval Estimate:
- Lower = 520 – 19 = $501
- Upper = 520 + 19 = $539
- (d) Interpretation: We are 95% confident that the true average amount that households will spend each month on food will lie between $501 and $539.