1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 7.4 Estimation of a Population Mean  is unknown  This section presents methods for estimating a population mean when the population standard deviation   is not known.

2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. The sample mean x is still the best point estimate of the population mean  . Best Point Estimate _

3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. When σ is unknown, we must use the Student t distribution instead of the normal distribution. Requires new parameter df = Degrees of Freedom Student t Distribution ( t-dist )

4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. The number of degrees of freedom (df) for a collection of sample data is defined as: “The number of sample values that can vary after certain restrictions have been imposed on all data values.” In this section: df = n – 1 Basically, since σ is unknown, a data point has to be “sacrificed” to make s. So all further calculations use n – 1 data points instead of n. Definition

5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using the Student t Distribution The t-score is similar to the z-score but applies for the t-dist instead of the z-dist. The same is true for probabilities and critical values. P(t < -1) t α (Area under curve)(Critical value) NOTE: The values depend on df 0 0 α (area)

6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Important Properties of the Student t Distribution 1.Has a symmetric bell shape similar to the z-dist 2.Has a wider distribution than that the z-dist 3.Mean μ = 0 4.S.D. σ > 1 (Note: σ varies with df) 5.As df gets larger, the t-dist approaches the z-dist

7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Student t Distributions for n = 3 and n = 12

8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. z-Distribution and t-Distribution Wider Spread df = 2 df = 100 As df increases, the t-dist approaches the z-dist Almost the same

9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. df = 2df = 3 df = 4 df = 5 df = 6df = 7df = 8 df = 20df = 50df = 100 Progression of t-dist with df

10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Choosing the Appropriate Distribution Use the normal (Z) distribution  known and normally distributed population or  known and n > 30 Use t distribution Methods of Ch. 7 do not apply Population is not normally distributed and n ≤ 30  not known and normally distributed population or  not known and n > 30

11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculating values from t-dist Stat → Calculators → T

12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculating values from t-dist Enter Degrees of Freedom (DF) and t-score

13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculating values from t-dist P(t<-1) = when df = 20

14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculating values from t-dist t α = when α = 0.05 df = 20

15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Margin of Error E for Estimate of  (σ unknown) Formula 7-6 where t  2 has n – 1 degrees of freedom. t  /2 = The t-value separating the right tail so it has an area of  /2

16 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. C.I. for the Estimate of μ (With σ Not Known)

17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Point estimate of µ : Margin of Error: Finding the Point Estimate and E from a C.I.

18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 Example: s Note:Same parameters as example used in Section : Etimating a population mean: σ known Using σ = 10 ( instead of s = 10.0 ) we found the 90% confidence interval: C.I. = (35.9, 40.9)

19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 Example: s Direct Computation: T Calculator (df = 41).0

20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 Example: s.0 Using StatCrunch Stat → T statistics → One Sample → with Summary

21 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 Example: s.0 Using StatCrunch Enter Parameters, click Next

22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 Example: s.0 Using StatCrunch Select Confidence Interval and enter Confidence Level, then click Calculate

23 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 Example: s.0 Using StatCrunch From the output, we find the Confidence interval is CI = (35.8, 41.0) Lower Limit Upper Limit Standard Error

24 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example: s If σ known Used σ = 10 to obtain 90% CI: If σ unknown Used s = 10.0 to obtain 90% CI: Notice: σ known yields a smaller CI (i.e. less uncertainty) Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 Results (35.8, 41.0) (35.9, 40.9)