Chapter 7 – Statistical Inference and Sampling Introduction to Business Statistics, 6e Kvanli, Pavur, Keeling Chapter 7 – Statistical Inference and Sampling Slides prepared by Jeff Heyl, Lincoln University ©2003 South-Western/Thomson Learning™

Simple Random Sampling All items in the population have the same probability of being selected Finite Population: To be sure that a simple random sample is obtained from a finite population the items should be numbered from 1 to N Nearly all statistical procedures require that a random sample is obtained

Estimation The population consists of every item of interest Population mean is µ and is generally not known The sample is randomly drawn from the population Sample values should be selected randomly, one at a time, from the population

Random Sampling and Estimation Figure 7.1 Population (mean = µ) Sample (mean = X) X estimates µ

Distribution for Everglo Bulb Lifetime | µ = 400 X  = 50 300 350 450 500 Figure 7.2

Sample Means Figure 7.3

Excel Histogram Frequency Histogram Class Limits Figure 7.4 8 7 6 5 4 3 2 1 377 and 384 and 391 and 398 and 405 and 412 and 419 and 426 and under 384 under 391 under 398 under 405 under 412 under 419 under 426 under 433 Class Limits Figure 7.4

Distribution of X The mean of the probability distribution for X = µX = µ Standard error of X = standard deviation of the probability distribution for X = X =  n

Normal Curves x X Population (mean = µ, standard deviation = ) Random sample (mean = X, standard deviation = s  = 50 x X = value from this population x = 50 10 X X follows a normal distribution, centered at µ with a standard deviation  / n µx = 400 Assumes the individual observations follow a normal distribution Figure 7.5

Central Limit Theorem When obtaining large samples (n > 30) from any population, the sample mean X will follow an approximate normal distribution What this means is that if you randomly sample a large population the X distribution will be approximately normal with a mean µ and a standard deviation (standard error) of x =  n

Distribution of X Figure 7.6  = 50 µ = 400 Population X µx = 400 = 15.81 50 10 (n = 50) = 7.07 (n = 20) = 11.18 20 (n = 100) = 5 100 Figure 7.6

Distribution of X Mean = µx = µ Standard deviation = x =  (standard error)  n

Assembly Time | 14 17 µ = 20 23 26 22  = 3 Area = P(X > 22) X = assembly line Figure 7.7

Assembly Time | 14 19.23 µx 20.77 22 x = .77 Area = P(X > 22) X Figure 7.8

Assembly Time | 19 µx = 20 21 Area = P(19 < X < 21) X Figure 7.9

Central Limit Theorem Figure 7.10 Uniform population µ = = 100 µ = = 100  = = 28.87 a + b 2 b - a 12 a = 50 µ = 100 b = 150 X (n = 2) (n = 5) (n = 30) µx By the CLT, µx = µ = 100 x = = = 5.27  n 28.87 30 Figure 7.10

Exponential population Central Limit Theorem µ = 100 =  X Exponential population (n = 2) (n = 5) (n = 30) µx By the CLT, µx = µ = 100 x = = = 18.26  n 100 30 Figure 7.11

Central Limit Theorem Figure 7.12 | µ X U-shaped population (n = 2)

Sampling Without Replacement Mean = µx = µ Standard deviation = x = • (standard error)  n N - n N - 1

Distribution of Sample Mean µx = 48,000 X = average income of 45 female managers x = = = (1267.11)(.935) = $1184.75  n N - n N - 1 8500 45 350 - 45 350 - 1 Observed value of X = $43,900 Figure 7.13

Confidence Intervals X µ = ?  known Figure 7.14

Confidence Intervals 3 x = = .6 minute 25 Area = P(X > 20) = .5 X = average of 25 assembly lines x = = .6 minute 3 25 Area = P(X > 20) = .5 µx = 20 Figure 7.15

Confidence Intervals Area = .475 -1.96 1.96 Total area = .95 Z -1.96 1.96 Total area = .95 Z Figure 7.16

Confidence for the Mean of a Normal Population ( known) P(-1.96  Z  1.96) = .95 Z = X - µ  / n P -1.96 ≤ ≤ 1.96 = .95 P X - 1.96 ≤ µ ≤ X + 1.96 = .95  n

Confidence for the Mean of a Normal Population ( known) (1 - ) • 100% Confidence Interval x - Z/2 , x + Z/2  n E = margin of error = Z/2  n

Confidence for the Mean of a Normal Population ( known) Area = .1 1.96 Z Area = .05 Area = .025 1.645 1.28 Figure 7.17

Excel Screens Figure 7.18

Excel Screens Figure 7.19

Excel Screens Figure 7.20

Confidence for the Mean of a Normal Population ( unknown) Student’s t Distribution Population variance unknown Degrees of freedom = n - 1 x - t/2, n - 1 to x + t/2, n - 1 s n

Student’s t Distribution t Standard normal, Z t curve with 20 df t curve with 10 df Figure 7.21

Confidence Interval Figure 7.22

Confidence Interval Figure 7.23

Selecting Necessary Sample Size Known  Sample size based on the level of accuracy required for the application Maximum error: E Used to determine the necessary sample size to provide the specified level of accuracy Specified in advance

Selecting Necessary Sample Size Known  E = Z/2  n n = Z/2 •  E 2

Selecting Necessary Sample Size Unknown  To obtain a rough approximation, ask someone who is familiar with the data to be collected: What do you think will be the highest value in the sample (H)? What will be the lowest value (L)?

Selecting Necessary Sample Size Unknown  Z/2 • s E 2   H - L 4

Other Sampling Procedures Population: the collection of all items about which we are interested Sampling Unit: a collection of elements selected from the population Cluster: a sampling unit that is a group of elements from the population, such as all adults in a particular city block Sampling frame: a list of population elements

Other Sampling Procedures Strata: are nonoverlapping subpopulations Sampling design: specifies the manner in which the sampling units are to be selected

Simple Random Sampling Population mean: µ Estimator: Estimated standard error of X: Approximate confidence interval: X = ∑x n  N - n N - 1 sx = • X ± Z/2sx

Systematic Sampling The sampling frame consists of N records The sample of n is obtained by sampling every kth record, where k is an integer approximately equal N/n The sampling frame should be ordered randomly

Stratified Sampling Stratified sampling obtains more information due to the homogenous nature of each strata Stratified sampling obtains a cross section of the entire population Obtain a mean within each strata as well as an estimate of 

Stratified Sampling Use the following notation: ni = sample size in stratum i Ni = number of elements in stratum i N = total population size = ∑Ni n = total sample size = ∑ni Xi = sample mean in stratum i si = sample standard deviation in stratum i

Stratified Sampling Population mean: µ Estimator: Estimated standard error of X: Approximate confidence interval: Xst = ∑NiXi N sx = ∑ st Ni 2 Ni - ni si ni Xst ± Z/2sx

Cluster Sampling Single-stage cluster sampling: randomly select a set of clusters for sampling Include all elements in the cluster in your sample Two-stage cluster sampling: randomly select a set of clusters for sampling Randomly select elements from each sampled cluster

Cluster Sampling Population mean: µ Estimator: ∑Ti Estimated standard error of Xc: Approximate confidence interval: Xc = ∑Ti ∑ni Xc ± Z/2sx c sx = M - m mMN2 ∑(Ti - Xcni)2 m - 1

obtain approximate confidence Confidence Interval Constructing a Confidence Interval for a Population Mean  known  unknown Use Table A-4 (Z) Use Table A-5 (t) Can use Table A-4 (Z) to obtain approximate confidence interval if n > 30 Figure 7.25