Difference between revisions of "AP Statistics Curriculum 2007 Estim L Mean"
(→Interval Estimation of a Population Mean) |
|||
| Line 6: | Line 6: | ||
===Interval Estimation of a Population Mean=== | ===Interval Estimation of a Population Mean=== | ||
For large samples, interval estimation of the population means (or Confidence intervals) are constructed as follows. Choose a confidence level <math>(1-\alpha)100%</math>, where <math>\alpha</math> is small (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.). Then a <math>(1-\alpha)100%</math> confidence interval for <math>\mu</math> will be | For large samples, interval estimation of the population means (or Confidence intervals) are constructed as follows. Choose a confidence level <math>(1-\alpha)100%</math>, where <math>\alpha</math> is small (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.). Then a <math>(1-\alpha)100%</math> confidence interval for <math>\mu</math> will be | ||
| − | : <math>CI(\ | + | : <math>CI(\alpha): \overline{x} \pm z_{\alpha\over 2} E,</math> |
* The '''margine of error''' E is defined as | * The '''margine of error''' E is defined as | ||
| Line 14: | Line 14: | ||
* The '''Standard Error''' of the estimate <math>\overline {x}</math> is defined by <math>SE(\overline {x}) = {1\over \sqrt{n} \sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}</math> | * The '''Standard Error''' of the estimate <math>\overline {x}</math> is defined by <math>SE(\overline {x}) = {1\over \sqrt{n} \sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}</math> | ||
| − | * <math>z_{\alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_Normal_Critical | critical value]] for a [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] distribution at <math>{\alpha\over 2}</math>. | + | * <math>z_{\alpha\over 2}</math> is the [[AP_Statistics_Curriculum_2007_Normal_Critical | critical value]] for a [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal]] distribution at <math>{\alpha\over 2}</math>. |
===Example=== | ===Example=== | ||
Revision as of 22:24, 2 February 2008
Contents
General Advance-Placement (AP) Statistics Curriculum - Estimating a Population Mean: Large Samples
Point Estimation of a Population Mean
For any process, the population mean may be estimated by a (large) sample average. That is the smaple average \(\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}\), constructed from a random sample of the procees {\(X_1, X_2, X_3, \cdots , X_n\)}, is an unbiased estimate of the population mean \(\mu\), if it exists! Note that the sample average may be susseptible to outliers.
Interval Estimation of a Population Mean
For large samples, interval estimation of the population means (or Confidence intervals) are constructed as follows. Choose a confidence level \((1-\alpha)100%\), where \(\alpha\) is small (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.). Then a \((1-\alpha)100%\) confidence interval for \(\mu\) will be \[CI(\alpha): \overline{x} \pm z_{\alpha\over 2} E,\]
- The margine of error E is defined as
\(E = \begin{cases}{\sigma\over\sqrt{n}},& \texttt{for-known}-\sigma,\\ {SE},& \texttt{for-unknown}-\sigma.\end{cases}\)
- The Standard Error of the estimate \(\overline {x}\) is defined by \(SE(\overline {x}) = {1\over \sqrt{n} \sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}\)
- \(z_{\alpha\over 2}\) is the critical value for a Standard Normal distribution at \({\alpha\over 2}\).
Example
Market researchers use the number of sentences per advertisement as a measure of readability for magazine advertisements. A random sample of the number of sentences found in 30 magazine advertisements is listed. Use this sample to find point estimate for the population mean \(\mu\).
| 16 | 9 | 14 | 11 | 17 | 12 | 99 | 18 | 13 | 12 | 5 | 9 | 17 | 6 | 11 | 17 | 18 | 20 | 6 | 14 | 7 | 11 | 12 | 5 | 18 | 6 | 4 | 13 | 11 | 12 |
A confidence interval estimate of \(\mu\) is a range of values used to estimate a population parameter (interval estimates are normally used more than point estimates because it is very unlikely that the sample mean would match exactly with the population mean) The interval estimate uses a margin of error about the point estimate. For example if you have a point estimate of 12. 25 with a margin of error of 1.75, then the interval estimate would be (10.5 to 14).
Before you find an interval estimate, you should first determine how confident you want to be that your interval estimate contains the population mean.
- 80% confidence (0.80), \(\alpha=0.1\), z = 1.28
- 90% confidence (0.90), \(\alpha=0.05\), z = 1.645
- 95% confidence (0.95), \(\alpha=0.025\), z = 1.96
- 99% confidence (0.99), \(\alpha=0.005\), z = 2.575
Known Variance
Suppose that we know the variance for the number of sentences per advertisement example above is known to be 256 (so the population standard deviation is \(\sigma=16\)).
- For \(\alpha=0.1\), the \(90% CI(\mu)\) is constructed by:
- For \(\alpha=0.05\), the \(90% CI(\mu)\) is constructed by:
- For \(\alpha=0.005\), the \(90% CI(\mu)\) is constructed by:
Notice the increase of the CI's (directly related to the decrease of \(\alpha\)) reflecting our choice for higher confidence.
Unknown variance
Suppose that we do not know the variance for the number of sentences per advertisement but use the sample variance 273 as an estimate (so the sample standard deviation is \(\sigma=16.54\)).
- For \(\alpha=0.1\), the \(90% CI(\mu)\) is constructed by:
- For \(\alpha=0.05\), the \(90% CI(\mu)\) is constructed by:
- For \(\alpha=0.005\), the \(90% CI(\mu)\) is constructed by:
Notice the increase of the CI's (directly related to the decrease of \(\alpha\)) reflecting our choice for higher confidence.
Hands-on activities
See the SOCR Confidence Interval Experiment.
References
- SOCR Home page: http://www.socr.ucla.edu
Translate this page:
