1 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 6-3 Estimating a Population Mean: Small Samples

2 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman If1) 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 Small Samples Assumptions

3 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Student t Distribution If the distribution of a population is essentially normal, then the distribution of t = x - µ s n

4 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Student t Distribution If the distribution of a population is essentially normal, then the distribution of  is essentially a Student t Distribution for all samples of size n.  is used to find critical values denoted by t = x - µ s n t  / 2

5 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Using the TI Calculator Find the “t” score using the InvT command as follows: 2 nd - Vars – then invT( percent, degree of freedom )

6 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition Degrees of Freedom ( df ) corresponds to the number of sample values that can vary after certain restrictions have imposed on all data values

7 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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

8 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 # Specific # so that x = a specific number Any # n = 10 df = = 9 Any # Any # Any # Any # Any # Any # Any #

9 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Margin of Error E for Estimate of  Based on an Unknown  and a Small Simple Random Sample from a Normally Distributed Population

10 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 where t  / 2 has n - 1 degrees of freedom n E = t   s 2

11 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Confidence Interval for the Estimate of E Based on an Unknown  and a Small Simple Random Sample from a Normally Distributed Population

12 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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

13 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 n s

14 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 n s t  /2 found using invT

15 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.

16 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Student t distribution with n = 3 Student t Distributions for n = 3 and n = 12 0 Student t distribution with n = 12 Standard normal distribution Figure 6-5

17 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 6-6 Using the Normal and t Distribution

18 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.)

19 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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

20 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 = t  / 2 = E = t  2 s = (2.201)(15,873) = 10, n 12

21 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 = t  / 2 = E = t  2 s = (2.201)(15,873) = 10,085.3 n 12 x - E < µ < x + E

22 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 = t  / 2 = E = t  2 s = (2.201)(15,873) = 10,085.3 n 12 26, ,085.3 < µ < 26, ,085.3 x - E < µ < x + E

23 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 = t  / 2 = E = t  2 s = (2.201)(15,873) = 10,085.3 n 12 26, ,085.3 < µ < 26, ,085.3 $16,141.7 < µ < $36,312.3 x - E < µ < x + E

24 Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 = t  / 2 = E = t  2 s = (2.201)(15,873) = 10,085.3 n 12 26, ,085.3 < µ < 26, ,085.3 x - E < µ < x + E We are 95% confident that this interval contains the average cost of repairing a Dodge Viper. $16,141.7 < µ < $36,312.3