Applied Statistics I
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1 Applied Statistics I Liang Zhang Department of Mathematics, University of Utah July 17, 2008 Liang Zhang (UofU) Applied Statistics I July 17, / 23
2 Large-Sample Confidence Intervals Liang Zhang (UofU) Applied Statistics I July 17, / 23
3 Large-Sample Confidence Intervals Proposition If n is sufficiently large, the standardized variable Z = X µ S/ n has approximately a standard normal distribution. This implies that x ± z α/2 s n is a large-sample confidence interval for µ with confidence level approximately 100(1 α)%. This formula is valid regardless of the shape of the population distribution. Liang Zhang (UofU) Applied Statistics I July 17, / 23
4 Large-Sample Confidence Intervals Liang Zhang (UofU) Applied Statistics I July 17, / 23
5 Large-Sample Confidence Intervals Example (a variant of Problem 16) The charge-to-tap time (min) for a carbon steel in one type of open hearth furnace was determined for each heat in a sample of size 46, resulting in a sample mean time of and a sample standard deviation of Calculate a 95% confidence interval for true average charge-to-tap time. Liang Zhang (UofU) Applied Statistics I July 17, / 23
6 Large-Sample Confidence Intervals Liang Zhang (UofU) Applied Statistics I July 17, / 23
7 Large-Sample Confidence Intervals Example (Problem 19) The article Limited Yield Estimation for Visual Defect Sources (IEEE Trans. on Semiconductor Manuf., 1997: 17-23) reported that, in a study of a particular wafer inspection process, 356 dies were examined by an inspection probe and 201 of these passed the probe. Assuming a stable process, calculate a 95% confidence interval for the proportion of all dies that pass the probe. Liang Zhang (UofU) Applied Statistics I July 17, / 23
8 Large-Sample Confidence Intervals Liang Zhang (UofU) Applied Statistics I July 17, / 23
9 Large-Sample Confidence Intervals Proposition A confidence interval for a population proportion p with confidence level approximately 100(1 α)% has lower confidence limit = ˆp + z2 α/2 2n z α/2 1 + (z 2 α/2 )/n ˆpˆq n + z2 α/2 4n 2 and upper confidence limit = ˆp + z2 α/2 2n + z α/2 1 + (z 2 α/2 )/n ˆpˆq n + z2 α/2 4n 2 Liang Zhang (UofU) Applied Statistics I July 17, / 23
10 Large-Sample Confidence Intervals Liang Zhang (UofU) Applied Statistics I July 17, / 23
11 Large-Sample Confidence Intervals Example (Problem 16) The charge-to-tap time (min) for a carbon steel in one type of open hearth furnace was determined for each heat in a sample of size 46, resulting in a sample mean time of and a sample standard deviation of Calculate a 95% upper confidence bound for true average charge-to-tap time. Liang Zhang (UofU) Applied Statistics I July 17, / 23
12 Large-Sample Confidence Intervals Liang Zhang (UofU) Applied Statistics I July 17, / 23
13 Large-Sample Confidence Intervals Example (Problem 19) The article Limited Yield Estimation for Visual Defect Sources (IEEE Trans. on Semiconductor Manuf., 1997: 17-23) reported that, in a study of a particular wafer inspection process, 356 dies were examined by an inspection probe and 201 of these passed the probe. Assuming a stable process, calculate a 95% lower confidence bound for the proportion of all dies that pass the probe. Liang Zhang (UofU) Applied Statistics I July 17, / 23
14 Large-Sample Confidence Intervals Liang Zhang (UofU) Applied Statistics I July 17, / 23
15 Large-Sample Confidence Intervals Proposition A large-sample upper confidence bound for µ is µ < x + z α s n and a large-sample lower confidence bound for µ is µ > x z α s n A one-sided confidence bound for p results from replacing z α/2 by z α and ± by either + or in the CI formula for p. In all cases the confidence level is approximately 100(1 α)% Liang Zhang (UofU) Applied Statistics I July 17, / 23
16 Liang Zhang (UofU) Applied Statistics I July 17, / 23
17 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time with X = 64.95, s = time Liang Zhang (UofU) Applied Statistics I July 17, / 23
18 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time with X = 64.95, s = time What is the 95% confidence interval for the population mean µ? Liang Zhang (UofU) Applied Statistics I July 17, / 23
19 Liang Zhang (UofU) Applied Statistics I July 17, / 23
20 Theorem Let X 1, X 2,..., X n be a random sample from a normal distribution with mean µ and variance σ 2, where µ and σ are unknown. The random variable T = X µ S/ n has a probability distribution called a t distribution with n 1 degrees of freedom (df). Here X is the sample mean and S is the sample standard deviation. Liang Zhang (UofU) Applied Statistics I July 17, / 23
21 Liang Zhang (UofU) Applied Statistics I July 17, / 23
22 Liang Zhang (UofU) Applied Statistics I July 17, / 23
23 Liang Zhang (UofU) Applied Statistics I July 17, / 23
24 Properties of t Distributions: Liang Zhang (UofU) Applied Statistics I July 17, / 23
25 Properties of t Distributions: Let t ν denote the density function curve for ν df. 1. t ν is governed by only one parameter ν, the number of degrees of freedom. Liang Zhang (UofU) Applied Statistics I July 17, / 23
26 Properties of t Distributions: Let t ν denote the density function curve for ν df. 1. t ν is governed by only one parameter ν, the number of degrees of freedom. 2. Each t ν curve is bell-shaped and centered at 0. Liang Zhang (UofU) Applied Statistics I July 17, / 23
27 Properties of t Distributions: Let t ν denote the density function curve for ν df. 1. t ν is governed by only one parameter ν, the number of degrees of freedom. 2. Each t ν curve is bell-shaped and centered at Each t ν curve is more spread out than the standard normal (z) curve. Liang Zhang (UofU) Applied Statistics I July 17, / 23
28 Properties of t Distributions: Let t ν denote the density function curve for ν df. 1. t ν is governed by only one parameter ν, the number of degrees of freedom. 2. Each t ν curve is bell-shaped and centered at Each t ν curve is more spread out than the standard normal (z) curve. 4. As ν increases, the spread of the corresponding t ν curve decreases. Liang Zhang (UofU) Applied Statistics I July 17, / 23
29 Properties of t Distributions: Let t ν denote the density function curve for ν df. 1. t ν is governed by only one parameter ν, the number of degrees of freedom. 2. Each t ν curve is bell-shaped and centered at Each t ν curve is more spread out than the standard normal (z) curve. 4. As ν increases, the spread of the corresponding t ν curve decreases. 5. As ν, the sequence of t ν curves approaches the standard normal curve (so the z curve is often called the t curve with df= ). Liang Zhang (UofU) Applied Statistics I July 17, / 23
30 Liang Zhang (UofU) Applied Statistics I July 17, / 23
31 Notation Let t α,ν = the number on the measurement axis for which the area under the t curve with ν df to the right of t α,ν is α; t α,ν is called a t critical value. Liang Zhang (UofU) Applied Statistics I July 17, / 23
32 Notation Let t α,ν = the number on the measurement axis for which the area under the t curve with ν df to the right of t α,ν is α; t α,ν is called a t critical value. Liang Zhang (UofU) Applied Statistics I July 17, / 23
33 Liang Zhang (UofU) Applied Statistics I July 17, / 23
34 Proposition Let x and s be the sample mean and sample standard deviation computed from the results of a random sample from a normal population with mean µ. Then a 100(1 α)% confidence interval for µ is ( x t α 2,n 1 or, more compactly, x ± t α n. An upper confidence bound for µ is 2,n 1 s s n, x + t α 2,n 1 x + t α,n 1 s n ) s n and replacing + by in this latter expression gives a lower confidence bound for µ, both with confidence level 100(1 α)%. Liang Zhang (UofU) Applied Statistics I July 17, / 23
35 Liang Zhang (UofU) Applied Statistics I July 17, / 23
36 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time with X = 64.95, s = time Liang Zhang (UofU) Applied Statistics I July 17, / 23
37 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time with X = 64.95, s = time What is the 95% confidence interval for the 11th component? Liang Zhang (UofU) Applied Statistics I July 17, / 23
38 Liang Zhang (UofU) Applied Statistics I July 17, / 23
39 Proposition A prediction interval (PI) for a single observation to be selected from a normal population distribution is x ± t α 2,n 1 s The prediction level is 100(1 α)% n Liang Zhang (UofU) Applied Statistics I July 17, / 23
40 Liang Zhang (UofU) Applied Statistics I July 17, / 23
41 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time with X = 64.95, s = time Liang Zhang (UofU) Applied Statistics I July 17, / 23
42 Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time with X = 64.95, s = time What is the 95% confidence interval such that at least 90% of the values in the population are inside this interval? Liang Zhang (UofU) Applied Statistics I July 17, / 23
43 Liang Zhang (UofU) Applied Statistics I July 17, / 23
44 Proposition A tolerance interval for capturing at least k% of the values in a normal population distribution with a confidence level 95%has the form x ± (tolerance critical value) s Liang Zhang (UofU) Applied Statistics I July 17, / 23
45 Proposition A tolerance interval for capturing at least k% of the values in a normal population distribution with a confidence level 95%has the form x ± (tolerance critical value) s The tolerance critical values for k = 90, 95, and 99 in combination with various sample sizes are given in Appendix Table A.6. Liang Zhang (UofU) Applied Statistics I July 17, / 23
46 Confidence Intervals for the Variance of a Normal Population Liang Zhang (UofU) Applied Statistics I July 17, / 23
47 Confidence Intervals for the Variance of a Normal Population Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time with X = 64.95, s = time Liang Zhang (UofU) Applied Statistics I July 17, / 23
48 Confidence Intervals for the Variance of a Normal Population Example (a variant of Problem 62, Ch5) The total time for manufacturing a certain component is known to have a normal distribution. However, the mean µ and variance σ 2 for the normal distribution are unknown. After an experiment in which we manufactured 10 components, we recorded the sample time which is given as follows: time with X = 64.95, s = time What is a 95% confidence for the population variance σ 2? Liang Zhang (UofU) Applied Statistics I July 17, / 23
49 Confidence Intervals for the Variance of a Normal Population Liang Zhang (UofU) Applied Statistics I July 17, / 23
50 Confidence Intervals for the Variance of a Normal Population Theorem Let X 1, X 2,..., X n be a random sample from a distribution with mean µ and variance σ 2. Then the random variable (n 1)S 2 σ 2 = (Xi X ) 2 has s chi-squared (χ 2 ) probability distribution with n 1 degrees of freedom (df). σ 2 Liang Zhang (UofU) Applied Statistics I July 17, / 23
51 Confidence Intervals for the Variance of a Normal Population Liang Zhang (UofU) Applied Statistics I July 17, / 23
52 Confidence Intervals for the Variance of a Normal Population Liang Zhang (UofU) Applied Statistics I July 17, / 23
53 Confidence Intervals for the Variance of a Normal Population Liang Zhang (UofU) Applied Statistics I July 17, / 23
54 Confidence Intervals for the Variance of a Normal Population Notation Let χ 2 α,ν, called a chi-squared critical value, denote the number on the measurement axis such that α of the area under the chi-squared curve with ν df lies to the right of χ 2 α,ν. Liang Zhang (UofU) Applied Statistics I July 17, / 23
55 Confidence Intervals for the Variance of a Normal Population Notation Let χ 2 α,ν, called a chi-squared critical value, denote the number on the measurement axis such that α of the area under the chi-squared curve with ν df lies to the right of χ 2 α,ν. Liang Zhang (UofU) Applied Statistics I July 17, / 23
56 Confidence Intervals for the Variance of a Normal Population Liang Zhang (UofU) Applied Statistics I July 17, / 23
57 Confidence Intervals for the Variance of a Normal Population Proposition A 100(1 α)% confidence interval for the variance σ 2 of a normal population has lower limit and upper limit (n 1)s 2 /χ 2 α 2,n 1 (n 1)s 2 /χ 2 1 α 2,n 1 A confidence interval for σ has lower and upper limits that are the square roots of the corresponding limits in the interval for σ 2. Liang Zhang (UofU) Applied Statistics I July 17, / 23
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