1 / 20

OA Cancelled due to technical difficulties… 

OA Cancelled due to technical difficulties… . Tonight’s HW is A2: 541(11b = PANIC and 11d) We will start using the books next Friday!.

alec-ingram
Download Presentation

OA Cancelled due to technical difficulties… 

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. OA Cancelled due to technical difficulties…  Tonight’s HW is A2: 541(11b = PANIC and 11d) We will start using the books next Friday!

  2. Chapter 23: Confidence Intervals for a Population MeanIn addition to using interval estimates for population proportions (p), we can also use confidence intervals to estimate the true value of a population mean (µ).

  3. The formula for a CI for a population mean is based on the sampling distribution of the sample mean . Recall that: • is approximately normally distributed when n > 30 or when the population is normal.

  4. Thus far, we have been using z-critical values from the normal distribution to make our confidence intervals.

  5. Now, remember that a z-score is calculated as • As long as is normally distributed, the distribution of z will also be normal. Why? • Since m, s, n are all constants, the shape shouldn’t change, only the center and spread.

  6. However, if we don’t know s(the population standard deviation) and we use s (the sample standard deviation) to estimate s, the distribution of isn’t quite normal, even if is approximately normal. So, we give this quantity a new name:

  7. If you have your book open it to the back and look at it now. You will get t-tables on the AP exam and can also find them in the back of your book.

  8. Since t is based on 2 variables (s and ) instead of just (like z), • the t-distributions have more variability than the z (normal) distribution. • However, as the sample size increases, s gets closer to sand the t-distributions get closer to the standard normal distribution (z). • The t-curves are very similar to normal curves, except that they are wider and defined by a number called the DEGREES OF FREEDOM. For this chapter, df = n – 1.

  9. Properties of the t-distributions: • The t-curve corresponding to any fixed number of degrees of freedom (df) is bell-shaped, symetricand centered at 0. • Each t-curve is more spread out than the z-curve (standard normal curve). • As the df increase, the spread of the corresponding t-curve decreases. • As the number of df increases, the t-curves get closerto the z-curve.

  10. T-curves are wider than the z-curve, so we must go further than 1.96 SD to capture 95% of the area. • To find out how far, we use a t-critical value from the t-table. • If df = 20 and you want 95% confidence, what t-critical value should you use? • Thus, to capture the middle 95% of the t-distribution with 20 df, you must go out 2.086stdev. 2.086

  11. 2.262 1.711 2.369 Find the t critical values for the following: • a. 95% confidence with n = 10 • b. 90% confidence with n = 25 • c. 99% confidence with n = 100 • When using the t-table and the df you want are not provided, round down to the nearest df given.

  12. Suppose that a machine is designed to produce bolts that have a diameter of 5 mm. Every hour a random sample of 15 bolts is selected and a 95% confidence interval for the mean diameter is constructed. If there is evidence that m ≠ 5, the machine is adjusted. In one particular sample, the mean diameter was 5.08 mm with a standard deviation of 0.11 mm. Calculate the interval and decide if you need to adjust the machine. • It is asking for a confidence interval, what should we do?

  13. That’s right, PANIC!

  14. P: We want to estimate the true mean bolt diameters that the machine is producing (mD). A: • We have a random sample of 15 bolts. • With a SRS, each bolt diameter is independent of the other. • Assuming there are at least 150 bolts produced each hour, our sample is less than 10% of the population. • LEN - …

  15. 3 ways to satisfy LEN: • Data comes from a normal population • Graphed data is unimodal and symmetric • n > 30 None of these are true but… LEN: It is reasonable to assume that bolt diameters produced by the machine are normally distributed.

  16. N: The conditions for a 95% 1-sample t interval have been met. DF = 14.

  17. I: =5.08 2.145(.11/√15) = (5.091, 5.1409) C: I am 95% confident that the interval from 5.091to 5.1409captures the true mean diameter of bolts. Does the machine need to be adjusted?

  18. Tonight’s HW: A2: 541(11b , d)& stay safe…

More Related