1 / 28

CHAPTER 2

CHAPTER 2. Statistical Inference 2.1 Estimation Confidence Interval Estimation for Mean and Proportion Determining Sample Size 2.2 Hypothesis Testing: Tests for one and two means Test for one and two proportions. Statistical Inference.

melody
Download Presentation

CHAPTER 2

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. CHAPTER 2 Statistical Inference 2.1 Estimation • Confidence Interval Estimation for Mean and Proportion • Determining Sample Size 2.2 Hypothesis Testing: • Tests for one and two means • Test for one and two proportions

  2. Statistical Inference • The field of statistical inference consist of those methods used to make decisions or to draw conclusions about a population. • These methods utilize the information contained in a sample from the population in drawing conclusions. • Statistical Inference may be divided into two major areas: parameter estimation and hypothesis testing. • To construct and interpret confidence interval estimates for the mean and the proportion. • How to determine the sample size necessary to develop a confidence interval for the mean or proportion In estimation, you will learn:

  3. Estimation • In interval estimation, an interval is constructed around the point estimate and it is stated that this interval is likely to contain the corresponding population parameter. • Each interval is constructed with regard to a given confidence level and is called a confidence interval. The confidence level associated with a confidence interval states how much confidence we have that this interval contains the true population parameter. The confidence level is denoted by . Lower Confidence Limit Upper Confidence Limit Point Estimate Width of confidence interval

  4. Confidence Intervals Confidence Intervals Population Proportion Population Mean σKnown σUnknown EQT 373

  5. Confidence Interval Estimates for Population Mean, μ

  6. Example If a random sample of size n=20 from a normal population with variance , and mean , construct a 95% confidence interval for the population mean, μ. Given n=20, and For 95% confident interval, we have From standard normal table: Solution:

  7. Example A publishing company has just published a new textbook. Before the company decides the price at which to sell this textbook, it wants to know the average price of all such textbooks in the market. The research department at the company took a sample of 36 comparable textbooks and collected the information on their prices. this information produced a mean price RM 70.50 for this sample. It is known that the standard deviation of the prices of all such textbooks is RM4.50. Construct a 90% confidence interval for the mean price of all such college textbooks.

  8. Solution: Given n=36, and For 90% confident interval, we have From standard normal table:

  9. Confidence Interval Estimates for the Difference between Two Population Mean,

  10. Example The scientist wondered whether there was a difference in the average daily intakes of dairy products between men and women. He took a sample of n =50 adult women and recorded their daily intakes of dairy products in grams per day. He did the same for adult men. A summary of his sample results is listed below. Construct a 95% confidence interval for the difference in the average daily intakes of daily products for men and women. Can you conclude that there is a difference in the average daily intakes of daily products for men and women?

  11. Solution: Hence, 95% CI, we have Thus, we conclude that there is a difference in the average daily intakes for men and women as ` .

  12. Confidence Interval Estimates for Population Proportion • The CI for p for n≥30

  13. Example According to the analysis of Women Magazine in June 2005, “Stress has become a common part of everyday life among working women in Malaysia. The demands of work, family and home place an increasing burden on average Malaysian women”. According to this poll, 40% of working women included in the survey indicated that they had a little amount of time to relax. The poll was based on a randomly selected of 1502 working women aged 30 and above. Construct a 95% confidence interval for the corresponding population proportion.

  14. Solution: Let p be the proportion of all working women age 30 and above, who have a limited amount of time to relax, and let be the corresponding sample proportion. From the given information, n=1502, Hence 95% CI Thus, we can state with 95% confidence that the proportion of all working women aged 30 and above who have a limited amount of time to relax is between 37.5% and 42.5%.

  15. Confidence Interval Estimates for the Differences between Two Population Proportion The CI for

  16. Example A researcher want to estimate the difference between the percentages of users of two toothpastes who will never switch to another toothpaste. In a sample of 500 users of Toothpaste A taken by this researcher, 100 said that the will never switch to another toothpaste. In another sample of 400 users of Toothpaste B taken by the same researcher, 68 said that they will never switch to another toothpaste. Construct a 97% confidence interval for the difference between the proportions of all users of the two toothpastes who will never switch.

  17. Solution: Toothpaste A : n1 = 500 and x1 = 100 Toothpaste B : n2 = 400and x2 = 68 97% confidence interval: Thus, with 97% confidence we can state that the difference between the proportions of all users of the two toothpastes who will never switch is between -0.026 and 0.086.

  18. Determining Sample Size Determining Sample Size For the Mean For the Proportion EQT 373

  19. Sampling Error The required sample size can be found to reach a desired margin of error (e) with a specified level of confidence (1 - ) The margin of error is also called sampling error • the amount of imprecision in the estimate of the population parameter • the amount added and subtracted to the point estimate to form the confidence interval EQT 373

  20. Determining Sample Size for population mean problems Determining Sample Size For the Mean Sampling error (margin of error)

  21. Example • If  = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence • So the required sample size is n = 220 (Always round up) EQT 373

  22. Determining Sample Size for population proportion problems Determining Sample Size For the Proportion Now solve for n to get EQT 373

  23. Example How large a sample would be necessary to estimate the true proportion defective in a large population within ±3%,with 95% confidence? (Assume a sample yields p = 0.12) For 95% confidence, we have Zα/2 = 1.96 e = 0.03; p= 0.12 Solution: So use n = 451

  24. Example A team of efficiency experts intends to use the mean of a randomsample of size n=150 to estimate the average mechanical aptitude of assembly-line workers in a large industry (as measured by a certain standardized test). If, based on experience, the efficiency experts can assume that for such data, what can they assert with probability 0.99 about the maximum error of their estimate? Substituting n=150, σ=6.2 and into the expression for the maximum error, we get Thus, the efficiency experts can assert with probability 0.99 that their error will be less than 1.30. Solution:

  25. Example A study is made to determine the proportion of voters in a sizable community who favor the construction of a nuclear power plant. If 140 of 400 voters selected at random favor the project and we use as an estimate of the actual proportion of all voters in the community who favor the project, what can we say with 99% confidence about the maximum error?

  26. Solution:

More Related