1 / 16

Confidence Interval for a Mean

Confidence Interval for a Mean. Confidence Interval for a Mean. Given A random sample of size n from a Normal population or a non Normal population where n is sufficiently large. A population at least 20 times the sample size n . Confidence Interval for a Mean. Result

zofia
Download Presentation

Confidence Interval for a Mean

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Confidence Interval for a Mean

  2. Confidence Interval for a Mean Given A random sample of size n from a Normal population or a non Normal population where n is sufficiently large. A population at least 20 times the sample size n.

  3. Confidence Interval for a Mean Result A confidence interval is given by where t* is the appropriate critical value for the T distribution with (n – 1) DF.

  4. Example Rolls (single rolls) of paper leave a factory with weights that are Normal with unknown mean. n = 8 rolls are randomly selected 1494.5 1483.8 1512.3 1507.0 1503.6 1504.5 1495.4 1521.5

  5. Example Rolls (single rolls) of paper leave a factory with weights that are Normal with unknown mean. n = 8 rolls are randomly selected, their sample mean weight is 1502.83 pounds; the sample standard deviation is 11.62 pounds. Determine a 95% confidence interval for the population (or true) mean weight.

  6. t* = 2.365 T7 Example 0.95 0.025 0.025

  7. Example Rolls (single rolls) of paper leave a factory with weights that are Normal with unknown mean. n = 8 rolls are randomly selected, their sample mean weight is 1502.83 pounds; the sample standard deviation is 11.62 pounds. Determine a 95% confidence interval for the population (or true) mean weight.

  8. Interpretation 1502.83  9.72 1502.83 – 9.72 = 1493.11 1502.83 + 9.72 = 1512.55 (1493.11, 1512.55) 95% of all samples* produce an interval that covers the true mean . We have the interval from one sample, chosen randomly. We are 95% confident that the mean weight of all rolls is between 1493.11 and 1512.55 pounds. *Provided sampling is done from a Normal distribution. Because the sample is small, if this requirement is not met, the confidence is not really 95%.

  9. TRUE / FALSE QUIZ 1. There’s a 95% confidence a roll is between 1493.11 and 1512.55 pounds. 1494.5 1483.8 1512.3 1507.0 1503.6 1504.5 1495.4 1521.5

  10. TRUE / FALSE QUIZ 1. There’s a 95% confidence a roll is between 1493.11 and 1512.55 pounds. 1494.5 1483.8 1512.31507.0 1503.6 1504.5 1495.4 1521.5 For this sample: 75%. 95% would be impossible (although closest would be “all” – which is clearly not the case).

  11. TRUE / FALSE QUIZ 1. There’s a 95% confidence a roll is between 1493.11 and 1512.55 pounds. FALSE: If you want to make this kind of statement, construct a prediction interval. One way of forming a 95% prediction interval is with the interval from the 2.5th to the 97.5th percentiles. (For a data set with 8 observations, this is tough. There are other ways…)

  12. Example Rolls (single rolls) of paper leave a factory with weights that are Normal with unknown mean. n = 8 rolls are randomly selected 1494.5 1483.8 1512.3 1507.0 1503.6 1504.5 1495.4 1521.5

  13. TRUE / FALSE QUIZ 2. There’s a 95% confidence the sample mean is between 1493.11 and 1512.55 pounds. FALSE. The sample mean definitely is between them; we can be 100% confident in that, because the sample mean centers the interval.

  14. TRUE / FALSE QUIZ 3. There’s a 95% confidence another random sample of rolls will have mean between 1493.11 and 1512.55 pounds. FALSE: The confidence interval estimates the population mean weight.

  15. TRUE / FALSE QUIZ 4. There’s a 95% probability that the mean weight of all rolls is between 1493.11 and 1512.55 pounds. FALSE (but in a way “closest” to true). The probability is either 0 or 1 – depending on what the population mean is.

  16. Interpretation What “95% confidence means” 95% of all samples produce an interval that covers the true mean . We have an interval from one sample, chosen randomly. Our interval either does or does not cover : in practice we just don’t know. We do know that the procedure works 95% of the time. Interpreting a 95% confidence interval We are 95% confident that the mean weight of all rolls is between 1493.11 and 1512.55 pounds.

More Related