The Normal Probability Distribution What is a distribution

The Normal Probability Distribution

What is a distribution? A collection of scores, values, arranged to indicate how common various values, or scores are. Mean (population, sample) Standard deviation (population, sample) Median Mode

Scores in our class

CHARACTERISTICS OF A NORMAL DISTRIBUTION Normal curve is symmetrical two halves identical - Tail Theoretically, curve extends to - infinity Tail Mean, median, and mode are equal Theoretically, curve extends to + infinity

AREAS UNDER THE NORMAL CURVE l About 68 percent of the area under the normal curve is within plus one and minus one standard deviation of the mean. This can be written as m ± 1 s. l About 95 percent of the area under the normal curve is within plus and minus two standard deviations of the mean, written m ± 2 s. l Practically all (99. 74 percent) of the area under the normal curve is within three standard deviations of the mean, written m ± 3 s.

Between: m ± 1 s 68. 26% m ± 2 s 95. 44% m± 3 s 99. 97% m-3 s m-2 s m-1 s m m+1 s m+2 s m+3 s

Normal Distributions with Equal Means but Different Standard Deviations. s = 3. 1 s = 3. 9 s = 5. 0 m = 20

Normal Probability Distributions with Different Means and Standard Deviations. m = 5, s = 3 m = 9, s = 6 m = 14, s = 10

What is this good for? ? • describes the data and how it clusters, arranges around a mean. • it’s good for us because it can allow us to make statistical inferences

CHARACTERISTICS OF A NORMAL PROBABILITY DISTRIBUTION l A normal distribution with a mean of 0 and a standard deviation of 1 is called the standard normal distribution l z value: The distance between a selected value, designated X, and the population mean m, divided by the population standard deviation, s. Disguised under z-score, normal scores, standardized score

What is it good for? Indicates how many standard deviations an observation is above/below the mean It’s good, because it allows us to compare observations from other normal distributions Is a 3. 00 GPA UNLV student as good as a 3. 00 GPA UCF student?

EXAMPLE 1 l The monthly incomes of recent high school graduates in a large corporation are normally distributed with a mean of $2, 000 and a standard deviation of $200. What is the z value for an income X of $2, 200? $1, 700? l For X = $2, 200 and since z = (X - m) / s, then z =.

EXAMPLE 1 (continued) l For X = $1, 700 and since z = (X - m)/s, then l A z value of +1. 0 indicates that the value of $2, 200 is ___ standard deviation ______ the mean of $2, 000. l A z value of – 1. 5 indicates that the value of $1, 700 is ____ standard deviation ______ the mean of $2, 000.

EXAMPLE 2 l The daily water usage person in Toledo, Ohio is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons. l About 68% of the daily water usage person in Toledo lies between what two values? l m ± 1 s = _______ l That is, about 68% of the daily usage person will lie between _________ gallons. l Similarly for 95% and 99%, the intervals will be _____________________.

POINT ESTIMATES l Point estimate: one number (called a point) that is used to estimate a population parameter. l Examples of point estimates are the sample mean, the sample standard deviation, the sample variance, the sample proportion, etc. EXAMPLE: The number of defective items produced by a machine was recorded for five randomly selected hours during a 40 -hour work week. The observed number of defectives were 12, 4, 7, 14, and 10. So the sample mean is ____. Thus a point estimate for the weekly mean number of defectives is 9. 4.

INTERVAL ESTIMATES l Interval Estimate: states the range within which a population parameter probably lies. l The interval within which a population parameter is expected to occur is called a confidence interval. l The two confidence intervals that are used extensively are the 95% and the 99%. l A 95%confidence interval means that about 95% of the similarly constructed intervals will contain the parameter being estimated.

INTERVAL ESTIMATES (continued) l Another interpretation of the 95% confidence interval is that 95% of the sample means for a specified sample size will lie within 1. 96 standard deviations of the hypothesized population mean. l For the 99% confidence interval, 99% of the sample means for a specified sample size will lie within 2. 58 standard deviations of the hypothesized population mean.

Determining Sample Size for Probability Samples Financial, Statistical, and Managerial Issues The larger the sample, the smaller the sampling error, but larger samples cost more. Budget Available Rules of Thumb

Typical Sample Sizes Number of subgroup analyses None/few Average Many Consumer research National Special population Business research* National Special population 200 -500 100 -500 20 -100 20 -50 500 -1000 200 -1000 50 -200 50 -100 1000 -2000 500 -1000 200 -500 100 -250

Sample Size Determination Sample size depends on Allowable Error/level of precision/ sampling error (E) Acceptable confidence in standard errors (Z) Population standard deviation ( )

Sample size determination Problem involving means: Sample Size (n) = Z 2 2 / E 2 where: Z = level of confidence expressed in standard errors = population standard deviation E = acceptable amount of sampling error

Sample size determination Problem involving proportions: Sample Size (n) = Z 2 [P(1 -P)] / E 2

Sampling Exercise Let us assume we have a population of 5 people whose names and ages are given below: Abe Bob Cara Don Emily 24 30 36 42 36

Average of all samples of size = 1 Abe 24 Bob 30 Cara Don 42 Emily 36 48 Average of all possible “size = 1” samples= 36

Average of all samples of size = 2 Abe, Bob (24+30)/2 = 27 Abe, Cara 30 Abe, Don 33 Bob, Cara 33 Abe, Emily 36 Bob, Don 36 Bob, Emily 39 Cara, Don 39 Cara, Emily 42 Don, Emily 45 Average of all possible “size = 2” samples= 36

Average of all samples of size = 3 Abe, Bob, Cara 30 Abe, Bob, Don 32 Abe, Bob, Emily 34 Abe, Cara, Don 34 Abe, Cara, Emily 36 Bob, Cara, Don 36 Bob, Cara, Emily 38 Abe, Don, Emily 38 Bob, Don, Emily 40 Cara, Don, Emily 42 Average of all possible “size = 3” samples= 36

Average of all samples of size = 4 Abe, Bob, Cara, Don 33 Abe, Bob, Cara, Emily Abe, Bob, Don, Emily Abe, Cara, Don, Emily Bob, Cara, Don, Emily 34. 5 36 37. 5 39 Average of all possible “size = 4” samples= 36

Average of all samples of size = 3 Abe, Bob, Cara, Don, Emily 36 Average of all possible “size = 5” samples= 36

What can be learned? What is the average of the sample for a given size? Does the mean of any individual sample equal to the population mean? Range of values for each sample size category?

Sampling Distribution Population distribution: A frequency distribution of all the elements of a population. Sample distribution: A frequency distribution of all the elements of an individual sample. Sampling distribution- a frequency distribution of the means of many samples.

Normal Distribution Central Limit Theorem - Central Limit Theorem— distribution of a large number of sample means or sample proportions will approximate a normal distribution, regardless of the distribution of the population from which they were drawn

The Standard Error of the Mean Applies to the standard deviation of a distribution of sample means. x = √ n

Sampling Distribution of the Proportion The Standard Error of the Distribution of Proportions Applies to the standard deviation of a distribution of sample proportions. Sp = √ P (1 -P) n where: Sp = standard error of sampling distribution proportion P = estimate of population proportion n = sample size

Sample size determination – adjusting for population size Make an adjustment in the sample size if the sample size is more than 5 percent of the size of the total population. Called the Finite Population Correction (FPC). x = √ n √ N-n N-1