Upcoming Schedule PSU Stat 2014

Transcription

1 Upcoming Schedule PSU Stat 014 Monday Tuesday Wednesday Thursday Friday Jan 6 Sec 7. Jan 7 Jan 8 Sec 7.3 Jan 9 Jan 10 Sec 7.4 Jan 13 Chapter 7 in a nutshell Jan 14 Jan 15 Chapter 7 test Jan 16 Jan 17 Final Review MLK Jr Day No School Jan 1 Monday schedule Jan Final Per 1-3 Jan 3 Final Per 4-6 Jan 4 Final Per 7,8 Final Review

2 Review Term Sample standard deviation Symbol s Population standard deviation s Sample Variance s Population Variance s Bluman, Chapter 7

3 7-4 Confidence Intervals for Variances and Standard Deviations When products that fit together (such as pipes) are manufactured, it is important to keep the variations of the diameters of the products as small as possible; otherwise, they will not fit together properly and will have to be scrapped. In the manufacture of medicines, the variance and standard deviation of the medication in the pills play an important role in making sure patients receive the proper dosage. For these reasons, confidence intervals for variances and standard deviations are necessary. Bluman, Chapter 7 3

4 Chi-Square Distributions The chi-square distribution must be used to calculate confidence intervals for variances and standard deviations. The chi-square variable is similar to the t variable in that its distribution is a family of curves based on the number of degrees of freedom. The symbol for chi-square is (Greek letter chi, pronounced ki ). A chi-square variable cannot be negative, and the distributions are skewed to the right. Bluman, Chapter 7 4

5 Chi-Square Distributions At about 100 degrees of freedom, the chi-square distribution becomes somewhat symmetric. The area under each chi-square distribution is equal to 1.00, or 100%. Bluman, Chapter 7 5

6 Formula for the Confidence Interval for a Variance; see page s right left n s n s, d.f. = n 1 Bluman, Chapter 7 6

7 Formula for the Confidence Interval for a Standard Deviation n 1 s 1 s n s, d.f. = n 1 right left Rounding Rule: Please refer to the bottom of page 388. Bluman, Chapter 7 7

8 Chapter 7 Confidence Intervals and Sample Size Section 7-4 Example 7-13 Page #387 Bluman, Chapter 7 8

9 Example 7-13: Using Table G right left Find the values for and for a 90% confidence interval when n = 5. right To find, subtract = Divide by to get To find, subtract to get left Bluman, Chapter 7 9

10 Example 7-13: Using Table G Use the 0.95 and 0.05 columns and the row corresponding to 4 d.f. in Table G. The value is ; the value is right left Bluman, Chapter 7 10

11 Using table G Generalized: 1) Calculate a which is the complement of the CL. ) chi square right: 1 α 3) chi square left: subtract the above value from 1 11

12 Confidence Interval for a Variance or Standard Deviation Rounding Rule When you are computing a confidence interval for a population variance or standard deviation by using raw data, round off to one more decimal places than the number of decimal places in the original data. When you are computing a confidence interval for a population variance or standard deviation by using a sample variance or standard deviation, round off to the same number of decimal places as given for the sample variance or standard deviation. Bluman, Chapter 7 1

13 Confidence Interval for a Variance or Standard Deviation Rounding Rule When you are computing a confidence interval for a population variance or standard deviation by using raw data, round off to one more decimal places than the number of decimal places in the original data. When you are computing a confidence interval for a population variance or standard deviation by using a sample variance or standard deviation, round off to the same number of decimal places as given for the sample variance or standard deviation. Bluman, Chapter 7 13

14 Chapter 7 Confidence Intervals and Sample Size Section 7-4 Example 7-14 Page #389 Bluman, Chapter 7 14

15 Example 7-14: Nicotine Content Find the 95% confidence interval for the variance and standard deviation of the nicotine content of cigarettes manufactured if a sample of 0 cigarettes has a standard deviation of 1.6 milligrams. right To find get left, subtract = Divide by to To find, subtract to get In Table G, the 0.05 and columns with the d.f. 19 row yield values of 3.85 and 8.907, respectively. Bluman, Chapter 7 15

16 Example 7-14: Nicotine Content 1 1 n s n s s right left s s 5.5 You can be 95% confident that the true variance for the nicotine content is between 1.5 and 5.5 milligrams. 1.5 s s.3 You can be 95% confident that the true standard deviation is between 1. and.3 milligrams. Bluman, Chapter 7 16

17 Chapter 7 Confidence Intervals and Sample Size Section 7-4 Example 7-15 Page #389 Bluman, Chapter 7 17

18 Example 7-15: Cost of Ski Lift Tickets Find the 90% confidence interval for the variance and standard deviation for the price in dollars of an adult single-day ski lift ticket. The data represent a selected sample of nationwide ski resorts. Assume the variable is normally distributed Using technology, we find the variance of the data is approximately s =8.; however use the VARS key to use the exact value in the formula. In Table G, the 0.05 and 0.95 columns with the d.f. 9 row yield values of and 3.35, respectively. Bluman, Chapter 7 18

19 Example 7-15: Cost of Ski Lift Tickets 1 1 n s n s s right left s < σ < 76.4 You can be 95% confident that the true variance for the cost of ski lift tickets is between 15.0 and

20 Example 7-15: Cost of Ski Lift Tickets 15.0 < σ < 76.4 To determine the interval for the standard deviation, don t use the rounded answers of the variation. Use the exact values < σ < < σ < 8.7 You can be 95% confident that the true standard deviation is between $3.9 and $8.7. Bluman, Chapter 7 0

21 Homework Sec 7.4 page 390 #5,8,9,11 Bluman, Chapter 7 1

22 Confidence Level left Column to use 90% % % % rigth Column to use Bluman, Chapter 7