a) Less than 0.66 b) Greater than 0.74 c) Between 0.64 and 0.76

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1 Example #7 According to a National Sleep Foundation Survey, 70% of adults takes more than 30 minutes to fall asleep at night. Assume that this percentage is true for the population of all U.S. adults. If a random sample of 200 adults is selected from this population, then determine the probability that the sample proportion, of adults who take more than 30 minutes to fall asleep will be:

2 a) Less than 0.66 b) Greater than 0.74 c) Between 0.64 and 0.76

3 a) Less than *0.30 pˆ P 0.70 p ˆ

4 a) Less than 0.66 pˆ pˆ pˆ

5 b) Greater than 0.74 pˆ pˆ

6 c) Between 0.64 and 0.76 pˆ low pˆ pˆ high

7 Example #8 According to the Princeton Research Group, 42% of U. S. drivers who live in the South drive at or below the speed limit. Assume this proportion is true for the population of all U.S. drivers living in the South. If a random sample of 300 drivers is selected from this population, then determine the probability that the sample proportion of drivers living in the South who drive at or below the speed limit is:

8 a) At most 39%. pˆ pˆ pˆ

9 b) At least 47% pˆ pˆ

10 c) Between 36% to 49% pˆ low pˆ pˆ high

11 Example #9 A large New York law firm states that 80% of their cases are settled out of court. What is the probability that out of the next 200 cases that:

12 a) At least 75% will be settled out of court? pˆ pˆ

13 b) At least 90% will be settled out of court? pˆ pˆ

14 c) At least 25% will not be settled out of court Is 75% WILL BE Same as part a

15 Example A National Survey conducted by The Social Science Research Center at Old Dominion University found that 56% of all drivers admitted they run red lights. Assume this proportion is true for the population of all U.S. drivers. If a random sample of 500 drivers is selected from this population, then determine the probability that the sample proportion of drivers who admit to running red lights is:

16 a) Less than 59% p P 0.59 p 0.56* p p

17 b) Between 53% and 58% pˆ low pˆ pˆ high

18 c) Greater than 61% pˆ pˆ

19 What would be the sampling error if we had selected a sample: d) With proportion = 59% of drivers who admit to running a red light. e) With proportion = 51% of drivers who admit to running a red light. f) Is it possible for the sampling error to be negative? pˆ pˆ

20 Example #11 According to a Gallup Poll, 60% of adults say they experience heartburn. Assume this proportion is true for the population of all adults. If a random sample of 300 adults is selected from this population, then determine the probability that:

21 a) Less than 172 adults within the sample experience heartburn.

22 a) Less than 172 pˆ pˆ pˆ

23 b) More than 191 pˆ pˆ

24 c) Between 164 to 198 pˆ low pˆ pˆ high

25 What would be the sampling error if we had selected a sample: d) With proportion pˆ = 59% of adults who experienced heartburn. e) With proportion pˆ = 51% of adults who experienced heartburn. f) Is it possible for the sampling error to be negative? Explain.

26 Example #12 The production characteristics for a small engine part are as follows: mean is 2.36", standard deviation is 0.04" and normally distributed. a) What percent of these parts fail to meet the engineering specifications of 2.35" plus or minus 0.1"?

27 a) 2.35 plus or minus 0.1 NORMAL x high x low

28 b) If sixteen of these parts are selected at random what is the probability that the average length is less than 2.35"?

29 b) less than 2.35" x 2.36 x x x

30 c) A quality control procedure requires that a random sample of size 4 be selected and that the sample mean be in the interval 2.32 to If not, then the case of parts is declared defective. What percent of these samples will fail to meet this test?

31 NORMAL c) interval not 2.32 to 2.38 x xlow x xhigh

32 Example #13 The birthrate for a county in Kentucky is stated to be 1.45 per women. Assume the population standard deviation is If a random sample of 400 women is selected from this county, what is the probability that the mean birthrate of the sample will fall between 1.20 and 1.70?

33 between 1.20 and 1.70 NORMAL x % x high x low

34 Example #14 (class room) A manufacturer of automobile batteries states their slow die battery has a mean life of 50 months with a standard deviation of 6 months.

35 If a consumer protection group randomly samples 49 of these batteries, what is the probability that the mean life of the consumer group s sample will be at most 52 months (assuming the manufacturer s claim is true)?

36 at most 52 x 50 x x x 50 52

37 Example #15(class room) A light bulb manufacturer states that its energy saver 60 watt light bulb has a mean life of 1850 hours with a standard deviation of 240 hours. If random samples of size 64 of these light bulbs are selected and tested, then within what interval (centered about the mean) would you expect 95% of these sample means to fall?

x 1850 x 240 64 30 97.5% = 0.975 x high 2.

38 x 1850 x % = x high 2.5% = % x low x

39 EXAM PREPARATION THERE ARE FOUR DISTRIBUTIONS

40 THERE ARE FOUR DISTRIBUTIONS NORMAL BINOMIAL SAMPLING OF THE MEANS SAMPLING OF THE PROPORTIONS

41 normalcdf ( low, high,, x x ) x n normalcdf ( low, high,, p p ) p pq n

42 Notes END

43 THERE ARE FOUR DISTRIBUTIONS NORMAL BINOMIAL SAMPLING OF THE MEAN SAMPLING OF THE PROPORTIONS

44 Procedure for all examples 1) Shade a diagram and label the mean, raw scores, z scores and the percentages of areas. 2) Show work for the use of all formulas. 3) Generate raw scores and percentages to at least two (2) decimal places. 4) Generate probabilities to at least four (4)* decimal places.

45 = probability Binomial- use boundaries = raw score

46 1. There are n identical trials. 2. The n identical trials are independent. 3. The outcome for each trial can be classified as either a success or a failure. 4. The probability of success, p, is the same for each trial.

47 Mean and Standard Deviation Binomial formulas np npq n = the number of trials p = the probability of a success in one trial q = the probability of a failure in one trial USE BOUNDARIES

48 Sampling Distributions Means normalcdf ( low, high,, x x ) Proportions x n normalcdf ( low, high,, p p p ) pq n

Binomial Distributions

Binomial Distributions Binomial Experiment The experiment is repeated for a fixed number of trials, where each trial is independent of the other trials There are only two possible outcomes of interest