This document contains examples and explanations of key concepts related to the normal distribution and probability, including: - Calculating probabilities for various z-scores under the standard normal distribution - Using the normal distribution to find probabilities for real-world scenarios involving cassette deck lifetimes, cornflake weights, and CEO ages - Properties of the sampling distribution of sample means, including how it approaches normality as sample size increases based on the Central Limit Theorem - Worked examples applying concepts like finding probabilities and cutoff scores for sample means and binomial experiments

1. Chapter 6: The Normal Distribution

2. Probability a. between z = 0 and z = 1.97

3. Probability b. between z=0 and z = -0.48

4. Probability c. to the left of z = 1.02

5. Probability d. to the right of z = 2.1

6. Probability e. to the left of -0.42

7. Probability f. area between z = 1.23 and z = 1.90

8. Probability g. between z =.24 and z = -1.12

9. The Standard Normal Distribution The standard normal distribution is the normal distribution with a mean of 0 and a standard deviation of 1. All normally distributed values can be transformed into the standard normal distribution by using the formula for the standard score: z = ___________ or z = ____________

10. Example Sunshine Stereo produces cassette decks that are guaranteed for 2 years. The cassette deck life is normally distributed with a mean of 2.3 years and a standard deviation of 0.4 years. We want to know the probability that a cassette deck will break down during the guarantee period of 2 years.

11. Example A factory has a machine that puts cornflakes into boxes. These boxes are advertised as having 20 oz. of cornflakes. Assume that the distributions of weights is normal with  =20 and  =1.5 oz. You pick a box at random from the product line. What is the probability that its weight will be between 19 and 21 oz.

12. Example The average age of CEOs is 56 years. Assume the variable is normally distributed. If the standard deviation is 4 years, find the probability that the age of a randomly selected CEO will be in the following range. Between 53 and 59 years old

13. Example The average age of CEOs is 56 years. Assume the variable is normally distributed. If the standard deviation is 4 years, find the probability that the age of a randomly selected CEO will be in the following range. Between 58 and 63 years old

14. Example The average age of CEOs is 56 years. Assume the variable is normally distributed. If the standard deviation is 4 years, find the probability that the age of a randomly selected CEO will be in the following range. Between 50 and 55 years old

15. When one must find the value of x, the following formula can be used: _________________________

16. Example: Quick Start Battery Company Quick Start Company makes 12-volt car batteries. After many years of product testing, they know that the average life of a Quick Start battery is normally distributed, with mean 45 months and standard deviation 8 months . If Quick Start guarantees a full refund on any battery that fails within the 36-month period after purchase, what percentage of their batteries will they expect to replace?

17. Example: Quick Start Battery Company Quick Start Company makes 12-volt car batteries. After many years of product testing, they know that the average life of a Quick Start battery is normally distributed, with mean 45 months and standard deviation 8 months . If Quick Start does not want to replace more than 10% of their batteries under full refund guarantee policy, for how long should they guarantee the batteries to the nearest month?

18. Example: To qualify for letter sorting, applicants are given a speed-reading test. The scores are normally distributed, with a mean of 80 and a standard deviation of 8. If only the top 15% of the applicants are selected, find the cutoff score.

19. Example To help students improve their reading, a school district decides to implement a reading program. It is to be administered to the bottom 5% of the students in the district, based on the scores on a reading achievement exam. If the average score for the students in the district is 122.6, find the cutoff score that will make a student eligible for the program. The standard deviation is 18. Assume the variable is normally distributed.

20. Example : If one-person household spends an average of $40 per week on groceries, find the maximum and minimum dollar amount spent per week for the middle 50% of one-person households. Assume that the standard deviation is $5 and the variable is normally distributed.

21. Distribution of Sample Means A sampling distribution of sample means is a distribution obtained by using the means computed from all possible random samples of a specific size taken from a population. Sampling error is the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population.

22. Properties of the Distribution of Sample Means The mean of the sample means will be the same as the population mean. The standard deviation of the sample means will be smaller than the standard deviation of the population, an it will be equal to the population standard deviation divided by the square root of the sample size. ____= __________ and ____= ___________

23. Central Limit Theorem As the sample size n increases without limit, the shape of the distribution of the sample means taken with replacement from a population with mean  and standard deviation  will approach a normal distribution. This distribution will have a mean  and a standard deviation  /√n

24. Example: Trout Lengths Suppose a team of biologists has been studying Pinedale Children’s Fishing Pond. Let x represent the length of a single trout take at random from the pond. This group of biologists has determined that x has a normal distribution with mean  =10.2 in. and standard deviation  =1.4 in. a) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?

25. b) What is the probability that the mean length of five trout taken at random is between 8 and 12 inches long?

26. Example: Height of Hospital Workers The average height of all the workers in a hospital is known to be 65 inches with a standard deviation of 2.3 inches. If a sample of 36 people is selected at random, what is the probability that the average height of these 36 people will be between 64 and 65.5 inches?

27. Example : The average amount of money that a depositor of the Second National City Bank has in an account is $5000 with a standard deviation of $650. A random sample of 36 accounts is taken. What is the probability that the average amount of money that these 36 depositors have in their accounts is between $4800 and $5300?

28. Example : The national average SAT score is 1019. Suppose that nothing is known about the shape of the distribution and that the standard is 100. If a random sample of 200 scores is selected and the sample mean is calculated to be 1050, would you be surprised?

29. Example The average purchase by a customer in a large novelty store is $4.00 with a standard deviation of $0.85. a) What is the probability that a single purchase is less than $3.70?

30. b) If 49 customers are selected at random, what is the probability that their average purchases will be less than $3.70?

31. The Normal Approximation to the Binomial Distribution

32. Continuity Correction Factor Binomial Normal X = 24 _________________ X > 24 _________________ X > 24 _________________ X < 24 _________________ X < 24 _________________

33. Mean and Standard Deviation Recall that the mean and standard deviation of a binomial experiment are  = ________  = ________ Use np>5 and nq>5 to test the appropriateness of the approximation

34. Example A theater owner has found that 5% of his patrons do not show up for the performance that they purchased tickets for. If the theater has 100 seats, find the probability that more than six patrons will not show up for the sold-out performance.

35. Example If the probability that a newborn child will be female is 50%, find the probability that in 100 births 55 or more will be females.

36. Example Forty-five percent of us have group O blood. Providence Memorial Hospital is conducting a blood drive because its supply of group O blood is low, and it needs 177 donors of group O blood. If 400 volunteers donate blood, estimate the probability that the number with group O blood is at least 177. Is the pool of 400 volunteers likely to be sufficient?