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## Introduction to Probability and Statistics for Engineering and the Sciences II

by: Alison Vandervort

82

0

26

# Introduction to Probability and Statistics for Engineering and the Sciences II STAT 428

Marketplace > Ohio State University > Statistics > STAT 428 > Introduction to Probability and Statistics for Engineering and the Sciences II
Alison Vandervort
OSU
GPA 3.58

Staff

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COURSE
PROF.
Staff
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Class Notes
PAGES
26
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KARMA
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This 26 page Class Notes was uploaded by Alison Vandervort on Monday September 21, 2015. The Class Notes belongs to STAT 428 at Ohio State University taught by Staff in Fall. Since its upload, it has received 82 views. For similar materials see /class/210006/stat-428-ohio-state-university in Statistics at Ohio State University.

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Date Created: 09/21/15
Confidence Interval Review Reading Sections 71 73 Spring 2009 Statistics 428 Confidence Interval Ideas Cls are ranges of values that are reasonable estimates guesses for population parameters Cls are ranges of parameter values that are consistent with the observed data Cls are constructed using probability distributions for unobserved sample statistics random variables The probability that an unobservedinterval where the limits are random variables captures the true population parameter value is the confidence of the interval 1d The probability that an unobservedinterval fails to capture the true population parameter value is the significance of the interval or Cls can be twosided half of the significance at high and low ends of the distribution onesided all of significance at one end of the distribution Spring 2009 Statistics 428 2 Strategy for Finding Confidence Intervals For 9 In General 1 Determine the distribution of the data 2 Choose a statistic 3 Using the data distribution find the distribution of the chosen statistic 4 Using the CDF for the statistic find limits c and d 5 Solve fore in the center of the interval and statistics a and b at the limits Spring 2009 Statistics 428 Confidence Interval Interpretation Calculated observed confidence intervals are interpreted without statements of probability or chance for the single calculated interval There is a 9 o the tr Istance falls between 7 If I repeated this sampling procedure 100 times I would expect 95 of the calculated intervals to capture the true average distance I am 95 confident that the true average distance is captured between 799 and 807 At 95 confidence the values between 799 and 807 are reasonable values for the true average distance Spring 2009 Statistics 428 Wider and Narrower For most confidence intervals To make a confidence interval wider Increase confidence 1 d decrease significance or be willing to make mistakes less often Decrease the sample size n To make a confidence interval narrower Decrease confidence 1 or increase significance or be willing to make mistakes more often Increase the sample size n Spring 2009 Statistics 428 Sample Size Rearrange the formula for a confidence interval to find the sample size necessary to limit the margin of error half the width of a symmetric confidence interval to a certain size at a given significance or For the CI for the mean u of a Normal distribution with known population standard deviation 0 2 o n 2 Spring 2009 Statistics 428 Confidence Intervals for Common Distributions Suppose X1 X2 Xn F 7 with mean a and variance 02 Calculate a l i or 100 con dence interval for the mean a o F NM 02 with known a An exact CI is given by 039 7 7 A K a X Zam aX Ada2 o F N a 02 with unknown 0 An exact CI is given by S X 2 n i X S t quote o F Bernoull p An exact Cl with an inexact con dence level is given by the Clopper Pearson Exact Interval or an approximate Cl is given by the Wilson Score Interval 4 1 232 222 gtIlt a A a gtilt 2052 7L 4172 p 2n ZaZ 7 2 22 22 02 1 1T 7 2 z i A 042 p 27L o F is unknown7 but n is large An approximate Cl is given by s S X tnil 1X t 2quotnl Practice Problem Suppose you have a circuit with two electrical devices eg two pumps a power supply and a circuit breaker Each electrical device uses energy according to a Uniform distribution with minimum wattsT1 and maximum wattsT iid X1X2 N UT 1T Because you don t want to blow too many circuits you would like to know the distribution of the total number of amperes that the devices use It turns out that the sum of two independent uniformly distributed RVs with the above limits has a triangular distribution Y X1 l X2 Y N T7 739 Spring 2009 Statistics 428 y T l x T 1 Z 7 fy 7 y T l l for T y 71 Note if you want to find the cdf you need two different cdfs one for each of the cases above The first is usual the second should be 05 the integral of the second expression OB TY 04 OD T1 T 141 y Using a single observation of YX1X2 Construct a 1 or100 o confidence interval for T Spring 2009 Statistics 428 9 Solution Using the strategy 1 Determine the distribution of the data Y N T 77 2 Choose a statistic We only have one observation so there s really only one choice of statistic Y 3 Using the data distribution find the distribution of the chosen statistic Y N T7 739 4 Using the CDF for the statistic find limits c and d Prc Y d1 a Put half the error below c and half above d PrY S C 042 PrY Z d 042 Use the properties of what we know must be a square triangle at the bottom and top of the distribution Area 12heightwidth 12width2 for right triangles as we have here Spring 2009 Statistics 428 10 The lower part The upper part PI39Y 3 052 PIG2d Dz2 width2 042 ilwidthf 052 gig P1 042 T1 dl2 a2 c PomE d MU w Put it together PrT 1 E Y T1 a1 oz 5 Solve for T in the center of the interval and statistics a and b in the limits PrY 1E T Y1 E1 oz Confidence Interval Y 1amp373Y1 amp Spring 2009 Statistics 428 11 Suppose you observe Y1O watts Calculate and interpret a 96 confidence interval for the average total watts T or 1 O96 004 ya 02 10102grg 101 02 92108 I am 96 confident that the true average power used in total by both of the two electrical devices is between 92 and 108 watts Would the width of this interval likely increase or decrease if we observed more than 100 instances of power use DECREASE Would the width of this interval increase or decrease if we increased the confidence to 98 INCREASE Spring 2009 Statistics 428 12 More Practice for Common Distributions Problems assigned for practice Problems assigned for homework Spring 2009 Statistics 428 Confidence Interval Review Reading Sections 71 73 Spring 2009 Statistics 428 Confidence Interval Ideas Cls are ranges of values that are reasonable estimates guesses for population parameters Cls are ranges of parameter values that are consistent with the observed data Cls are constructed using probability distributions for unobserved sample statistics random variables The probability that an unobservedinterval where the limits are random variables captures the true population parameter value is the confidence of the interval 1d The probability that an unobservedinterval fails to capture the true population parameter value is the significance of the interval or Cls can be twosided half of the significance at high and low ends of the distribution onesided all of significance at one end of the distribution Spring 2009 Statistics 428 2 Strategy for Finding Confidence Intervals For 9 In General 1 Determine the distribution of the data 2 Choose a statistic 3 Using the data distribution find the distribution of the chosen statistic 4 Using the CDF for the statistic find limits c and d 5 Solve fore in the center of the interval and statistics a and b at the limits Spring 2009 Statistics 428 Confidence Interval Interpretation Calculated observed confidence intervals are interpreted without statements of probability or chance for the single calculated interval There is a 9 o the tr Istance falls between 7 If I repeated this sampling procedure 100 times I would expect 95 of the calculated intervals to capture the true average distance I am 95 confident that the true average distance is captured between 799 and 807 At 95 confidence the values between 799 and 807 are reasonable values for the true average distance Spring 2009 Statistics 428 Wider and Narrower For most confidence intervals To make a confidence interval wider Increase confidence 1 d decrease significance or be willing to make mistakes less often Decrease the sample size n To make a confidence interval narrower Decrease confidence 1 or increase significance or be willing to make mistakes more often Increase the sample size n Spring 2009 Statistics 428 Sample Size Rearrange the formula for a confidence interval to find the sample size necessary to limit the margin of error half the width of a symmetric confidence interval to a certain size at a given significance or For the CI for the mean u of a Normal distribution with known population standard deviation 0 2 o n 2 Spring 2009 Statistics 428 Confidence Intervals for Common Distributions Suppose X1 X2 Xn F 7 with mean a and variance 02 Calculate a l i or 100 con dence interval for the mean a o F NM 02 with known a An exact CI is given by 039 7 7 A K a X Zam aX Ada2 o F N a 02 with unknown 0 An exact CI is given by S X 2 n i X S t quote o F Bernoull p An exact Cl with an inexact con dence level is given by the Clopper Pearson Exact Interval or an approximate Cl is given by the Wilson Score Interval 4 1 232 222 gtIlt a A a gtilt 2052 7L 4172 p 2n ZaZ 7 2 22 22 02 1 1T 7 2 z i A 042 p 27L o F is unknown7 but n is large An approximate Cl is given by s S X tnil 1X t 2quotnl Practice Problem Suppose you have a circuit with two electrical devices eg two pumps a power supply and a circuit breaker Each electrical device uses energy according to a Uniform distribution with minimum wattsT1 and maximum wattsT iid X1X2 N UT 1T Because you don t want to blow too many circuits you would like to know the distribution of the total number of amperes that the devices use It turns out that the sum of two independent uniformly distributed RVs with the above limits has a triangular distribution Y X1 l X2 Y N T7 739 Spring 2009 Statistics 428 y T l x T 1 Z 7 fy 7 y T l l for T y 71 Note if you want to find the cdf you need two different cdfs one for each of the cases above The first is usual the second should be 05 the integral of the second expression OB TY 04 OD T1 T 141 y Using a single observation of YX1X2 Construct a 1 or100 o confidence interval for T Spring 2009 Statistics 428 9 Solution Using the strategy 1 Determine the distribution of the data Y N T 77 2 Choose a statistic We only have one observation so there s really only one choice of statistic Y 3 Using the data distribution find the distribution of the chosen statistic Y N T7 739 4 Using the CDF for the statistic find limits c and d Prc Y d1 a Put half the error below c and half above d PrY S C 042 PrY Z d 042 Use the properties of what we know must be a square triangle at the bottom and top of the distribution Area 12heightwidth 12width2 for right triangles as we have here Spring 2009 Statistics 428 10 The lower part The upper part PI39Y 3 052 PIG2d Dz2 width2 042 ilwidthf 052 gig P1 042 T1 dl2 a2 c PomE d MU w Put it together PrT 1 E Y T1 a1 oz 5 Solve for T in the center of the interval and statistics a and b in the limits PrY 1E T Y1 E1 oz Confidence Interval Y 1amp373Y1 amp Spring 2009 Statistics 428 11 Suppose you observe Y1O watts Calculate and interpret a 96 confidence interval for the average total watts T or 1 O96 004 ya 02 10102grg 101 02 92108 I am 96 confident that the true average power used in total by both of the two electrical devices is between 92 and 108 watts Would the width of this interval likely increase or decrease if we observed more than 100 instances of power use DECREASE Would the width of this interval increase or decrease if we increased the confidence to 98 INCREASE Spring 2009 Statistics 428 12 More Practice for Common Distributions Problems assigned for practice Problems assigned for homework Spring 2009 Statistics 428

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