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ELEMENTARY Section 6-3 STATISTICS Estimating a Population Mean: Small Samples EIGHTH Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION MARIO F. TRIOLA 1 Small Samples Assumptions If 1) n 30 2) The sample is a simple random sample. 3) The sample is from a normally distributed population. Case 1 ( is known): Largely unrealistic; Use methods from 6-2 Case 2 (is unknown): Use Student t distribution Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 2 Student t Distribution If the distribution of a population is essentially normal, then the distribution of t = x-µ s n Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3 Student t Distribution If the distribution of a population is essentially normal, then the distribution of t = x-µ s n is essentially a Student t Distribution for all samples of size n. is used to find critical values denoted by Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman t/ 2 4 Using the TI Calculator Find the “t” score using the InvT command as follows: 2nd - Vars – then invT( percent, degree of freedom ) Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5 Definition Degrees of Freedom (df ) corresponds to the number of sample values that can vary after certain restrictions have imposed on all data values Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6 Definition Degrees of Freedom (df ) corresponds to the number of sample values that can vary after certain restrictions have imposed on all data values df = n - 1 in this section Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7 Definition Degrees of Freedom (df ) = n - 1 corresponds to the number of sample values that can vary after certain restrictions have imposed on all data values Any Any Any Any Any # # # # # n = 10 Any Any # # Any Any Specific # # # df = 10 - 1 = 9 so that x = a specific number Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8 Margin of Error E for Estimate of Based on an Unknown and a Small Simple Random Sample from a Normally Distributed Population Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9 Margin of Error E for Estimate of Based on an Unknown and a Small Simple Random Sample from a Normally Distributed Population Formula 6-2 E = t / s 2 n where t/ 2 has n - 1 degrees of freedom Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10 Confidence Interval for the Estimate of E Based on an Unknown and a Small Simple Random Sample from a Normally Distributed Population Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11 Confidence Interval for the Estimate of E Based on an Unknown and a Small Simple Random Sample from a Normally Distributed Population x-E <µ< x +E Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12 Confidence Interval for the Estimate of E Based on an Unknown and a Small Simple Random Sample from a Normally Distributed Population x-E <µ< x +E where E = t/2 s n Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 13 Confidence Interval for the Estimate of E Based on an Unknown and a Small Simple Random Sample from a Normally Distributed Population x-E <µ< x +E where E = t/2 s n t/2 found using invT Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14 Properties of the Student t Distribution There is a different distribution for different sample sizes It has the same general bell shape as the normal curve but greater variability because of small samples. The t distribution has a mean of t = 0 The standard deviation of the t distribution varies with the sample size and is greater than 1 As the sample size (n) gets larger, the t distribution gets closer to the normal distribution. Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15 Student t Distributions for n = 3 and n = 12 Student t Standard normal distribution distribution with n = 12 Student t distribution with n = 3 Figure 6-5 0 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16 Using the Normal and t Distribution Figure 6-6 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 17 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 18 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873 = 0.05 /2 = 0.025 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 19 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873 E = t/2 s = (2.201)(15,873) = 10,085.29 n 12 = 0.05 /2 = 0.025 t/2 = 2.201 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 20 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873 = 0.05 /2 = 0.025 t/2 = 2.201 E = t/2 s = (2.201)(15,873) = 10,085.3 12 n x -E <µ< x +E Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 21 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873 = 0.05 /2 = 0.025 t/2 = 2.201 E = t/2 s = (2.201)(15,873) = 10,085.3 n x -E <µ< 26,227 - 10,085.3 < µ < 12 x +E 26,227 + 10,085.3 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 22 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873 = 0.05 /2 = 0.025 t/2 = 2.201 E = t/2 s = (2.201)(15,873) = 10,085.3 n x -E <µ< 26,227 - 10,085.3 < µ < $16,141.7 < µ < 12 x +E 26,227 + 10,085.3 $36,312.3 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 23 Example: A study of 12 Dodge Vipers involved in collisions resulted in repairs averaging $26,227 and a standard deviation of $15,873. Find the 95% interval estimate of , the mean repair cost for all Dodge Vipers involved in collisions. (The 12 cars’ distribution appears to be bell-shaped.) x = 26,227 s = 15,873 = 0.05 /2 = 0.025 t/2 = 2.201 E = t/2 s = (2.201)(15,873) = 10,085.3 n x -E <µ< 26,227 - 10,085.3 < µ < $16,141.7 < µ < 12 x +E 26,227 + 10,085.3 $36,312.3 We are 95% confident that this interval contains the average cost of repairing a Dodge Viper. Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 24