Need for Sampling. Very large populations Destructive testing Continuous production process


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1 Chapter 4 Sampling and Estimation
2 Need for Sampling Very large populations Destructive testing Continuous production process The objective of sampling is to draw a valid inference about a population. 4
3 Sample Design Sampling Plan a description of the approach that will be used to obtain samples from a population Objectives Target population Population frame Method of sampling Operational procedures for data collection Statistical tools for analysis 43
4 Sampling Methods Subjective Judgment sampling Convenience sampling Probabilistic Simple random sampling every subset of a given size has an equal chance of being selected 44
5 Excel Data Analysis Tool Sampling Excel menu > Tools > Data Analysis > Sampling Specify input range of data Choose sampling method Select output option 45
6 Other Sampling Methods Systematic sampling Stratified sampling Cluster sampling Sampling from a continuous process 46
7 Errors in Sampling Nonsampling error Poor sample design Sampling (statistical) error Depends on sample size Tradeoff between cost of sampling and accuracy of estimates obtained by sampling 47
8 Estimation Estimation assessing the value of a population parameter using sample data. Point estimate a single number used to estimate a population parameter Confidence intervals a range of values between which a population parameter is believed to be along with the probability that the interval correctly estimates the true population parameter 48
9 Common Point Estimates 49
10 Example 410
11 Theoretical Issues Unbiased estimator one for which the expected value equals the population parameter it is intended to estimate The sample variance is an unbiased estimator for the population variance s n i1 x i x x n 1 n i1 i N 411
12 Interval Estimates Range within which we believe the true population parameter falls Example: Gallup poll percentage of voters favoring a candidate is 56% with a 3% margin of error. Interval estimate is [53%, 59%] 41
13 Confidence Intervals Confidence interval (CI) an interval estimated that specifies the likelihood that the interval contains the true population parameter Level of confidence (1 ) the probability that the CI contains the true population parameter, usually expressed as a percentage (90%, 95%, 99% are most common). 413
14 Sampling Distribution of the Mean Theory 414
15 Interval Estimate Containing the True Population Mean 415
16 Interval Estimate Not Containing the True Population Mean 416
17 Confidence Interval for the Mean Known A 100(1 )% CI is: x z / (/n) z / may be found from Table A.1 or using the Excel function NORMSINV(1/) 417
18 Sampling From Finite Populations When n > 0.05N, use a correction factor in computing the standard error: x n N N n
19 Key Observations x z / (/n) As the confidence level (1  ) increases, the width of the confidence interval also increases. As the sample size increases, the width of the confidence interval decreases. 419
20 Confidence Interval for the Mean, Unknown A 100(1 )% CI is: x t /,n1 (s/n) t /,n1 is the value from a tdistribution with n1 degrees of freedom, from Table A. or the Excel function TINV(, n1) 40
21 Relationship Between Normal Distribution and tdistribution The tdistribution yields larger confidence intervals for smaller sample sizes. 41
22 Data Analysis, Fall 014 Confidence Intervals for a Proportion Sample proportion: p= x/n x = number in sample having desired characteristic n = sample size The sampling distribution of p has mean and variance (1 )/n When n and n(1 ) are at least 5, the sampling distribution of p approach a normal distribution 4
23 Data Analysis, Fall 014 Confidence Intervals for Proportions A 100(1 )% CI is: p z / p(1 n p) PHSta tool is available under Confidence Intervals option 43
24 Sampling Distribution of s The sample standard deviation, s, is a point estimate for the population standard deviation, The sampling distribution of s has a chisquare ( ) distribution with n1 df See Table A.3 CHIDIST(x, deg_freedom) returns probability to the right of x CHIINV(probability, deg_freedom) returns the value of x for a specified righttail probability 44
25 Confidence Intervals for the Variance A 100(1 )% CI is: ( n 1) s n1, /, ( n 1) s n1,1 / Note the difference in the denominators! 45
26 Confidence Intervals for Population Total s N n A 100(1 )% CI is: N x t n1,/ N n N 1 46
27 Confidence Intervals and Decision Making Required weight for a soap product is 64 ounces. A sample of 30 boxes found a mean of 63.8 and standard deviation of A 95% CI is [63.43, 64.1]. What conclusion can you reach? What if the standard deviation was 0.46 and the CI is [64.65, 63.99]? 47
28 Confidence Intervals and Sample Size CI for the mean, known Sample size needed for halfwidth of at most E is n (z / ) ( )/E CI for a proportion Sample size needed for halfwidth of at most E is ( z / ) (1 ) n E Use p as an estimate of or 0.5 for the most conservative estimate 48
29 Additional Types of Confidence Intervals Difference in means Independent samples with unequal variances Independent samples with equal variances Paired samples Difference in proportions 49
30 Confidence Intervals for Differences Between Means Population 1 Population Mean 1 Standard deviation 1 Point estimate x 1 x Sample size n 1 n Point estimate for the difference in means, 1, is given by x 1 x 430
31 Independent Samples With Unequal Variances 431 A 100(1 )% CI is: x 1 x (t /, df* ) 1 1 n s n s 1 ) / ( 1 ) / ( n n s n n s n s n s df* = Fractional values rounded down
32 Example In the Accounting Professionals Excel file, find a 95 percent confidence interval for the difference in years of service between males and females. 43
33 Calculations s 1 = 4.39 and n 1 = 14 (females), s = 8.39 and n = 13 (males) df* = 17.81, so use 17 as the degrees of freedom or ,
34 Independent Samples With Equal Variances A 100(1 )% CI is: x 1 x (t /, n1 + n ) s p 1 n 1 1 n s p ( n 1 1) s n 1 1 ( n n 1) s where s p is a common pooled standard deviation. Must assume the variances of the two populations are equal. 434
35 Example: Accounting Professionals Data s p or 14.87,
36 Paired Samples A 100(1 )% CI is: D (t n1,/ ) s D /n D i = difference for each pair of observations D = average of differences s D n i1 ( D i n 1 D) PHSta tool available in the Confidence Intervals menu 436
37 Example Pile Foundation A 95% CI for the average difference between the actual and estimated pile lengths is 437
38 Calculations D = 6.38 s D = % CI is 6.38 ± 1.96(10.31/ 311) = 6.38 ± or [5.34, 7.56] 438
39 Data Analysis, Fall 014 Differences Between Proportions A 100(1 )% CI is: p 1 p z / p 1(1 p 1) p (1 p n n 1 ) Applies when n i p i and n i (1 p i ) are greater than
40 Example In the Accounting Professionals Excel file, the proportion of females having a CPA is 8/14 = 0.57, while the proportion of males having a CPA is 6/13 = A 95 percent confidence interval for the difference in proportions between females and males is 440
41 Probability Intervals A 100(1 )% probability interval for a random variable X is any interval [a,b] such that P(a X b) = 1 Do not confuse a confidence interval with a probability interval; confidence intervals are probability intervals for sampling distributions, not for the distribution of the random variable. 441
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