IS 310 Business Statistic s CSU Long Beach IS 310 – Business

IS 310 Business Statistic s CSU Long Beach IS 310 – Business Statistics 1

Interval Estimation In chapter 7, we studied how to estimate a population parameter with a sample statistic. We used a point estimate as a single value. To increase the level of confidence in estimation, we use a range of values (rather than a single value) as the estimate of a population parameter. Let’s take a real-life example. Suppose, someone asks you how long it takes to go from CSULB to LAX. What would be a more reliable estimate – 30 minutes or between 25 and 40 minutes? If you use 30 minutes as estimate, you are using a point estimate. On the other hand, if you use between 25 and 40 minutes as estimate, you are using an interval estimation . IS 310 – Business Statistics 2

Interval Estimation Interval estimation uses a range of values. The width of the range indicates the level of confidence. The narrower the range, lower the confidence. Wider the range, higher the confidence. Once a confidence level is specified, the interval estimate can be calculated using the formula, 8.1, in the book. If one wants 95% confidence, the interval estimate is known as 95% Confidence Interval. For a 90% confidence, the interval is known as 90% Confidence Interval. IS 310 – Business Statistics 3

Interval Estimation In this chapter, we will cover the following: Interval Estimate for a Population Mean (known σ) Interval Estimate for a Population Mean (unknown σ) Determining Size of a Sample Interval Estimate for a Population Proportion IS 310 – Business Statistics 4

Margin of Error and the Interval Estimate The The general general form form of of an an interval interval estimate estimate of of aa population population mean mean is is x Margin of Error IS 310 – Business Statistics 5

Margin of Error The Margin of Error implies both Confidence level and Amount of error IS 310 – Business Statistics 6

Interval Estimate of a Population Mean: Known Interval Estimate of x z /2 n where: x is the sample mean 1 - is the confidence coefficient z /2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution is the population standard deviation n is the sample size IS 310 – Business Statistics 7

Interval Estimate for Population Mean (Known σ) Example: Lloyd’s Department Store wants to determine a 95% confidence interval on the average amount spent by all of its customers (population mean). Lloyd took a sample of 100 customers and found that the average amount spent is 82 (this is sample mean or x‾ ). Lloyd assumes that the population standard deviation ( σ ) is 20. IS 310 – Business Statistics 8

Interval Estimation for Population Mean (Known σ ) Using Formula 8.1, the 95% confidence interval is: x z (σ/ n) 0.05 0.025 82 1.96 (20/ 100) 82 3.92 78.08 to 85.92 We are 95% sure that the average amount spent by all Lloyd customers is between 78.08 and 85.92. IS 310 – Business Statistics 9

Sample Problem Problem # 5 (10-Page 306; 11-Page 315) Given: x 24.80 n 49 σ 5 confidence level 95% 0.05 or 5% a. b. Margin of Error z x (σ / n) /2 1.96 (5/ 49) 1.4 95% Confidence Interval is: x Margin of Error 24.8 1.4 23.4, 26.2 It means that we are 95% confident that the average amount spent by all customers for dinner at the Atlanta restaurant is between 23.40 and 26.20 IS 310 – Business Statistics 10

Interval Estimation of a Population Mean: Unknown If an estimate of the population standard deviation cannot be developed prior to sampling, we use the sample standard deviation estimatecase. . This is thes tounknown In this case, the interval estimate for is based on the t distribution. (We’ll assume for now that the population is normally distributed.) IS 310 – Business Statistics 11

t Distribution The The tt distribution distribution is is aa family family of of similar similar probability probability distributions. distributions. A A specific specific tt distribution distribution depends depends on on aa parameter parameter known known as as the the degrees degrees of of freedom freedom. Degrees Degrees of of freedom freedom refer refer to to the the sample sample size size minus minus 1 1 IS 310 – Business Statistics 12

t Distribution A A tt distribution distribution with with more more degrees degrees of of freedom freedom has has less less dispersion. dispersion. As As the the number number of of degrees degrees of of freedom freedom increases, increases, the the difference difference between between the the tt distribution distribution and and the the standard standard normal normal probability probability distribution distribution becomes becomes smaller smaller and and smaller. smaller. IS 310 – Business Statistics 13

t Distribution t distribution (20 degrees of freedom) Standard normal distribution t distribution (10 degrees of freedom) z, t 0 IS 310 – Business Statistics 14

t Distribution Degrees Area in Upper Tail of Freedom .20 .10 .05 .025 .01 .005 . . . . . . . 50 .849 1.299 1.676 2.009 2.403 2.678 60 .848 1.296 1.671 2.000 2.390 2.660 80 .846 1.292 1.664 1.990 2.374 2.639 100 .845 1.290 1.660 1.984 2.364 2.626 .842 1.282 1.645 1.960 2.326 2.576 Standard normal z values IS 310 – Business Statistics 15

Interval Estimation of a Population Mean: Unknown Interval Estimate x t /2 s n where: 1 - the confidence coefficient t /2 the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s the sample standard deviation IS 310 – Business Statistics 16

Interval Estimate for Population Mean (Unknown σ) Example: We want to estimate the mean credit card debt of all customers in the U.S. with a 95% confidence. A sample of 70 households provided the credit card balances shown in Table 8.3. The sample standard deviation (s) is 4007. Using formula 8.2, x t (s/ n) 0.05 0.025 9312 1.995 (4007/ 70) IS 310 – Business Statistics 17

Interval Estimate (Continued) 9312 955 8357 to 10,267 We are 95% confident that the average credit card balance of all customers in the U.S. are between 8,357 and 10,267. IS 310 – Business Statistics 18

Summary of Interval Estimation Procedures for a Population Mean Can the Yes population standard deviation be assumed known ? Known Case Use x z / 2 n IS 310 – Business Statistics No Use the sample standard deviation s to estimate s Use s Unknown x t / 2 Case n 19

Sample Problem Problem # 16 (10-Page 315; 11-Page 324) Given: x 49 n 100 s 8.5 Confidence Level 90% a. At 90% confidence, the Margin of Error is: t b. . (s/ n) 1.660 x (8.5/10) 1.411 /2 The 90% Confidence Interval is: x 1.411 49 1.411 47.59, 50.41 It means that we are 90% confident that the average hours of flying for Continental pilots is between 47.59 and 50.41 IS 310 – Business Statistics 20

Sample Size for an Interval Estimate of a Population Mean Let Let E E the the desired desired margin margin of of error. error. E E is is the the amount amount added added to to and and subtracted subtracted from from the the point point estimate estimate to to obtain obtain an an interval interval estimate. estimate. IS 310 – Business Statistics 21

Sample Size for an Interval Estimate of a Population Mean Margin of Error E z /2 n Necessary Sample Size ( z / 2 ) 2 2 n E2 IS 310 – Business Statistics 22

Determination of Sample Size Example Problem (10-Page 317; 11-Page 326): We want to know the average daily rental rate of all midsize automobiles in the U.S. We want our estimate with a margin of error of 2 and a 95% level of confidence. What should be the size of our sample? Given for this problem: E 2, 0.05, σ 9.65 Using Formula 8.3, 2 2 2 n (z ) (σ ) / E 0.025 89.43 The sample size needs to be at least 90. IS 310 – Business Statistics 23

Sample Problem Problem #26 (10-Page 318; 11-Page 327) Given: µ 2.41 σ 0.15 Margin of Error 0.07 Confidence Level 95% Margin of Error z . (σ / n) /2 2 2 2 0.07 1.96 (0.15/ n) n (1.96) (0.15) / 0.07) 17.63 or 18 A sample size of 18 is recommended for The Cincinnati Enquirer for its study IS 310 – Business Statistics 24

Interval Estimation of a Population Proportion The The general general form form of of an an interval interval estimate estimate of of aa population population proportion proportion is is p Margin of Error IS 310 – Business Statistics 25

Interval Estimation of a Population Proportion The of The sampling sampling distribution distribution p of plays plays aa key key role role in in computing computing the the margin margin of of error error for for this this interval interval estimate. estimate. The of The sampling sampling distribution distribution p of can can be be approximated approximated by by aa normal normal distribution distribution whenever whenever np np 5 5 and and n n(1 (1 –– p p)) 5. 5. IS 310 – Business Statistics 26

Interval Estimation of a Population Proportion Normal Approximation of Sampling Distributionp of Sampling p(1 p) p distribution n p of /2 1 - of all p values z / 2 p IS 310 – Business Statistics p /2 p z / 2 p 27

Interval Estimation of a Population Proportion Interval Estimate p z / 2 where: p (1 p ) n 1 - is the confidence coefficient z /2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution p is the sample proportion IS 310 – Business Statistics 28

Interval Estimate for a Population Proportion Example (10-Page 320; 11-Page 329): We want to estimate the proportion of all women golfers in the U.S. who are satisfied with the availability of tee times with a 95% confidence level. We take a sample of 900 women golfers and find that 396 are satisfied with the tee times. Given: p 396/900 0.44 0.05 The 95% confidence interval is: p z [p (1 – p)]/n 0.44 1.96 [0.44 (1 – 0.44)]/900 0.44 0.0324 0.4076 to 0.4724 or 40.76% to 47.24% We are 95% confident that the proportion of all women golfers who are satisfied with tee times is between 40.76% and 47.24%. IS 310 – Business Statistics 29

Sample Problem #38 (10-Page 323; 11-Page 332)) a. b. Point estimate of the proportion of companies that fell short of estimates 29/162 0.179 Given : p 104/162 0.642 Margin of error z [(p (1 – p ) / n)] 0.025 1.96 [(0.642) (1 – 0.642) / 162] 0.0738 IS 310 – Business Statistics 30

Sample Problem (contd) 95% confidence interval: 0.642 0.0738 0.5682 and 0.7158 We are 95% confident that the proportion of companies that beat estimates of their profits is between 56.82% and 71.58%. 2 2 c. Required sample size, n [(1.96) (0.642) (0.358)]/(0.05) 353.18 use 354 IS 310 – Business Statistics 31

End of Chapter 8 IS 310 – Business Statistics 32