Lecture 4: Estimating Population Mean - Confidence Intervals PDF

# Lecture 4: Estimating the Population Mean µ using Confidence Intervals (A,B) ## Introduction - We have a population of size N. - Population mean u is unknown. - We want to estimate u using the sample mean Xn obtained from a random sample. ## Definitions **Point Estimate:** An estimate of a po...

# Lecture 4: Estimating the Population Mean µ using Confidence Intervals (A,B) ## Introduction - We have a population of size N. - Population mean u is unknown. - We want to estimate u using the sample mean Xn obtained from a random sample. ## Definitions **Point Estimate:** An estimate of a population parameter given by a single number **Examples:** - Xn (sample mean) is a point estimate for u (population mean). - s (sample standard deviation) is a point estimate for σ (population standard deviation). ## Point Estimate Errors - There is some error when we estimate µ by Xn. - This error is represented by: X- µ = Std error = s / √n - n is the size of the sample - s is the standard deviation of the population. ## Confidence Intervals - A confidence interval (CI) for µ is an entire interval of plausible values for µ. - It's defined by a confidence level, e.g., 95%, 99%, etc. - Using Xn (our point estimate) as the center, we determine an interval of values for μ, (A,B), at the specified confidence level. ## Desirable Characteristics of Confidence Intervals - **Small confidence intervals:** we want to hone-in on the unknown mean μ. - **Large confidence levels:** we want to be more confident that the true population mean lies within the interval. ## Tradeoff between Confidence Level and Interval Size - As we try to increase the confidence level, the confidence interval gets wider. ## Case A: σ is known We want to create an interval estimate for the parameter µ when σ is known. ### Assumptions 1. **Normality:** The underlying population X is normally distributed, or large samples (n≥30) are taken. This ensures X is approximately normal due to the Central Limit Theorem. 2. **σ Known:** The population standard deviation σ is known. This is usually unrealistic as we are trying to estimate the population mean. 3. **Simple Random Sample:** The sample is a simple random sample. 4. **Independence:** The data values are independent. ### Formula for Confidence Interval The symmetric (1-α) CI for u is given by: Xn ± Za/2 * σ / √n Where: - Xn is the sample mean - Za/2 is the Z-score with area of α/2 to the right of it, under the Z curve. - σ is the population standard deviation - n is the sample size ### Example Let's consider an example with N = 250 students in a class. We are interested in the average weight of the students. - We take a random sample of n = 40 students. - The known population standard deviation σ = 6. - The sample mean X40 = 165 lbs. - We want to find a 90% confidence interval to estimate the population mean µ. #### Steps 1. **Confidence Level & Za/2:** We want a 90% confidence interval, so α = 0.1 and Za/2 = 1.645 (from the Z-table). 2. **Calculate Margin of Error:** E = Za/2 * σ / √n = 1.645 * 6 / √40 = 1.56 3. **Confidence Interval:** (Xn - E, Xn + E) = (165 - 1.56, 165 + 1.56) = (163.44, 166.56) #### Interpretation We are 90% confident that the true population mean weight lies in the interval (163.44, 166.56). ## Importance of the Interpretation - The 95% CI is presented as "We are 95% confident that the true parameter u lies in this interval. - This means that in repeated sampling, 95% of the intervals computed would include the population parameter µ. ## Example Continued Let's continue with the example from above and consider: 1. **60% confidence interval** 2. **98% confidence interval** 3. **Margin of sampling error for the 98% confidence interval** 4. **Length of the 98% confidence interval** The steps are similar to the steps for the 90% confidence interval but using different values for Za/2 and the confidence level. ## Conclusion - By understanding confidence intervals, we can obtain a range of plausible values for the unknown population mean. - This information is crucial for making informed decisions and drawing reliable conclusions in a variety of fields.