How To Find The Z Score, Confidence Interval, and Margin of Error for a Population Mean

Introduction to Confidence Intervals

• Discussion on how to find a confidence interval for a population mean.

• Importance of understanding z scores in relation to confidence intervals.

• Explanation of the margin of error (error bound for the mean).

Example Problem 1: Physics Class Test Scores

• Given Data:

  • Average test scores are normally distributed.

  • Standard deviation ($\sigma$) = 5.4.

  • Sample size ($n$) = 50.

  • Sample mean ($\bar{x}$) = 79.

Part A: Constructing a 95% Confidence Interval

• Objective: Estimate population mean using a confidence interval based on given data.

• Confidence Level: 95%

• Confidence Interval Formula:

  • $CI = \bar{x} \pm z_{\alpha} \frac{\sigma}{\sqrt{n}}$

  • Lower Bound: $\bar{x} - EBM$

  • Upper Bound: $\bar{x} + EBM$

• Error Bound for the Mean (EBM):

  • $EBM = z_{\alpha} \frac{\sigma}{\sqrt{n}}$

• Determining $z_{\alpha}$:

  • $z_{\alpha}$ corresponds to the area that is left of it in a standard normal distribution.

  • Area left of $z_{\alpha}$: $\frac{0.95 + 1}{2} = 0.975$

  • From z tables, finding $z$ for area 0.975 gives: $z_{\alpha} = 1.96$.

• Calculating Standard Deviation of Sample Mean:

  • $\frac{\sigma}{\sqrt{n}} = \frac{5.4}{\sqrt{50}} \approx 0.7636$.

• Calculating EBM:

  • $EBM = 1.96 \times 0.7636 = 1.4968$.

• Constructing Confidence Interval:

  • Lower Limit: $79 - 1.4968 = 77.5032$.

  • Upper Limit: $79 + 1.4968 = 80.4968$.

• Final 95% Confidence Interval:

  • $CI = (77.5032, 80.4968)$.

  • Interpretation: We are 95% confident the population mean is between 77.5032 and 80.4968.

Part B: Value of Margin of Error

• Margin of Error = EBM:

  • $EBM = 1.4968$.

Example Problem 2: Chemistry Class Test Scores

• Given Data:

  • Standard deviation ($\sigma$) = 6.5.

  • Sample size ($n$) = 100.

  • Sample mean ($\bar{x}$) = 82.

Part A: Constructing a 90% Confidence Interval

• Objective: To find a 90% confidence interval for population mean.

• Confidence Level: 90%

• Confidence Interval Formula:

  • $CI = \bar{x} \pm z_{\alpha} \frac{\sigma}{\sqrt{n}}$

• Determining $z_{\alpha}$:

  • Area left of $z_{\alpha}$: $\frac{0.90 + 1}{2} = 0.95$.

  • From the z score table: approximate $z=1.645$ (average of two surrounding values).

• Calculating Standard Deviation of Sample Mean:

  • $\frac{\sigma}{\sqrt{n}} = \frac{6.5}{\sqrt{100}} = 0.65$.

• Calculating EBM:

  • $EBM = z_{\alpha} \times \frac{\sigma}{\sqrt{n}} = 1.645 \times 0.65 = 1.06925$.

• Constructing Confidence Interval:

  • Lower Limit: $82 - 1.06925 = 80.93075$.

  • Upper Limit: $82 + 1.06925 = 83.06925$.

• Final 90% Confidence Interval:

  • $CI = (80.93075, 83.06925)$.

  • Interpretation: We are 90% confident that the population mean score is between 80.93075 and 83.06925.

Part B: Value of Margin of Error

• Margin of Error (EBM):

  • $EBM = 1.06925$.

Differences Between Confidence Interval and Confidence Level

• Confidence Interval (CI):

  • Range of values we expect the population mean to be within a certain probability.

  • Expressed as: $CI = (\bar{x} - EBM, \bar{x} + EBM)$.

• Confidence Level (CL):

  • Area under the curve where we expect the population mean to be located.

  • Example: For a 90% CL, $CL = 0.90$.

  • Relationship: $CL = 1 - \alpha$ where $\alpha$ is the area outside the CI.

Effects on Margin of Error

• Increasing Confidence Level:

  • A higher confidence level results in a longer confidence interval, thus increasing the margin of error (EBM).

• Increasing Sample Size:

  • Higher sample size decreases the margin of error because the standard error reduces as $n$ increases (since $(\sqrt{n})$ is in the denominator). Thus, the relationship is inverse.

Summary

• Confidence intervals provide estimates of population parameters with specified confidence levels.

• Techniques to compute confidence intervals involve critical values and the standard error, demonstrating the interplay between statistical concepts and sample data.

• The margin of error offers insight into the precision of those estimates and can be adjusted by manipulating confidence levels or sample sizes.

Confidence intervals (CI) are used to estimate a population mean. They are calculated using the formula $CI = \bar{x} \pm z<em>{\alpha} \frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $\sigma$ is the standard deviation, $n$ is the sample size, and $z</em>{\alpha}$ is the critical z-score corresponding to the desired confidence level. The term $z_{\alpha} \frac{\sigma}{\sqrt{n}}$ is known as the Error Bound for the Mean (EBM), or margin of error. We are 95% confident that the population mean is between 77.5032 and 80.4968, and for the second problem, we are 90% confident that the population mean score is between 80.93075 and 83.06925.

Confidence Level (CL) represents the area under the curve where the population mean is expected, while Confidence Interval (CI) is the range of values for that expectation. An increase in the confidence level leads to a larger EBM, resulting in a wider CI. Conversely, increasing the sample size reduces the EBM, leading to a narrower, more precise CI. Confidence intervals provide estimates of population parameters with specified confidence levels, offering insights into the precision of these estimates.