6. - the B-school Binomial Formula p = probability of success, q = 1- p = probability of failure, r = number of successes, n = total number of trials, µ = np = mean. = = standard deviation.
9. - the B-school Binomial Distribution – A Graphical Exploration Draw probability histograms of the binomial distribution for (a) n = 10 and p = 0.1, 0.3, 0.5, 0.7, 0.9 (b) p = 0.4 and n = 5, 10, 30
10. - the B-school Binomial Distribution – A Graphical Exploration The binomial probability histogram for n = 10 and p = 0.1,0.3,0.5,0.7,0.9
11. - the B-school Binomial Distribution – A Graphical Exploration The binomial probability histogram for p = 0.4 and n = 5,10,30
12.
13.
14.
15.
16. - the B-school Standard Normal Probability Distribution - Example Jarrid Medical, is developing a compact kidney dialysis machine but is having trouble controlling the variability of the rate at which fluid moves through the device. Medical standards require that the hourly flow be 4 litres, plus or minus, 0.1 litre, 80% of the time. Testing the prototype has revealed that 68% of the time the hourly flow is within 0.08 litres of 4.02 litres. Does the prototype satisfy medical standards?
17. - the B-school Standard Normal Probability Distribution – Example (Solution) Solution: = 4.02, = 0.08 Thus the prototype does not satisfy the medical requirements. (required rate of flow)
18.
19. - the B-school Central Limit Theorem If the random variables x 1 , x 2 ,…..x N are independent and identically distributed with mean µ and standard deviation , then is normally distributed with mean and standard deviation .
20. - the B-school Central Limit Theorem – Exercise If the weights of individual packets of 2 minute noodles varies according to a normal distribution with a mean of 85.8 grams and a standard deviation of 1.9 grams a. Describe the distribution that will describe the mean weight of a simple random sample of 5 such packets of noodles. Such a sample of 5 packets of noodles in a multi- pack is labeled “average weight of contents: 85 grams” b. Determine the proportion of such multi-packs with an average weight within 1 gram of the claimed average weight. c. Determine the proportion of such multi-packs with an average weight less than what is claimed.
21. - the B-school Central Limit Theorem – Exercise (solution) a. The mean weight is normally distributed with a mean of 85.8 grams and a standard deviation of b. c. (use NORMSDIST)
22. ESTIMATION POINT ESTIMATE INTERVAL ESTIMATE A single number is used to estimate an unknown population parameter. e.g our current data of MBA enrolments indicates that 20% of the students who will enrol next session will be women. Range of values used to estimate a population parameter. e.g our current data of MBA enrolments indicates that 15% to 24 % of the students who will enrol next session will be women.
23.
24. - the B-school Point Estimates - Example The National Bank of Lincon is trying to determine the number of tellers available during the lunch rush on Fridays. The bank has collected data on the number of people who entered the bank during the last 3 months on Friday from 11 A.M to 1 P.M. Using the data below, find the point estimates of the mean and standard deviation of the population from which the sample was drawn. 242 275 289 306 342 385 279 245 269 305 294 328
28. - the B-school Interval Estimates for Population Mean Seven homemakers were randomly sampled and it was determined that the distances they walked in their housework had an average of 39.2 miles per week and a sample standard deviation of 3.2 miles per week. Construct a 95% confidence interval for the population mean.
29. - the B-school Interval Estimates for Population Mean Solution: Sample size = n = 7 (we use t distribution since n ≤ 30 ) Degrees of freedom = 6. Sample mean = = 39.2 miles. Sample sd = = 3.2 miles (estimate of population sd) Standard error = x = = 3.2/2.645 = 1.209 t value (from t table under column 0.05, 6 df,use TINV) = 2.447 = 39.2 ± 2.447 * 1.209 = 39.2 ± 2.9596 = (36.240,42.160) miles
30. - the B-school Interval Estimates for Population Proportion A quality control inspector collected a random sample of 500 tubes of toothpaste from the production line and found that 41 of them had leaks from the tail end. Construct a 90% confidence interval for the percentage of all toothpaste tubes that had leakage.
31. - the B-school Interval Estimates for Population Proportion Solution: Sample size = 500 Point estimate of population proportion = = 41/500 = 0.082 Confidence level of p = 0.90 = 1 - α . α = 0.10 Standard error = critical z value = Z /2 (use NORMSINV(0.05)) = 1.645 = 0.082 ± 1.645* 0.0122 = 0.082 ±0.0201 = (0.062,0.1022) tubes Approx 6.2% – 10.2% of the tubes will have leakages.
32. - the B-school Determining Sample Size The university is considering raising tution to improve school facilities and they want to determine what percentage of students favour the increase. The university needs to be 90% confident that the percentage has been estimated to within 2% of the true value. How large a sample is needed to guarantee this accuracy regardless of the true percentage?
33. - the B-school Determining Sample Size Solution: z value for 90% confidence level is = 1.645 (NORMSINV(0.05)) Standard error = pq/n = 0.00014884 n = pq/0.00014884 The largest value of n will be obtained when pq is largest i.e when p = q = 0.5 n = 0.5*0.5/ 0.00014884 = 1680
![PROBABILITY DISTRIBUTIONS ,[object Object],[object Object],[object Object],[object Object],The Rationale for Using Probability Theory](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)