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)

25 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 7.5 Estimation of a Population Variance This section presents methods for estimating a population variance   and standard deviation .

26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. The sample variance s 2 is the best point estimate of the population variance   Best Point Estimate of  

27 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. The sample standard deviation s is the best point estimate of the population standard deviation  Best Point Estimate of 

28 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Pronounced “Chi-squared” Also dependent on the number degrees of freedom df. The  2 Distribution (  2 -dist )

29 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Properties of the  2 Distribution Chi-Square Distribution Use StatCrunch to Calculate values (similar to z-dist and t-dist) Chi-Square Distribution for df = 10 and df = 20 1.The chi-square distribution is not symmetric, unlike the z-dist and t-dist. 2.The values can be zero or positive, they are nonnegative. 3.Dependent on the Degrees of Freedom: df = n – 1

30 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculating values from  2 -dist Stat → Calculators → Chi-Squared

31 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculating values from  2 -dist Enter Degrees of Freedom DF and parameters ( same procedure as with t-dist ) P(  2 < 10)= when df = 10

32 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Find the 90% left and right critical values (  2 L and  2 R ) of the  2 -dist when df = 20 Example: Need to calculate values when the left/right areas are 0.05 ( i.e. α/2 )  2 L =  2 R =

33 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. The  2 -distribution is used for calculating the Confidence Interval of the Variance σ 2 Take the square-root of the values to get the Confidence Interval of the Standard Deviation σ ( This is why we call it  2 instead of  ) Important Note!!

34 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Confidence Interval for Estimating a Population Variance Note:Left and Right Critical values on opposite sides

35 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Confidence Interval for Estimating a Population Standard Deviation Note:Left and Right Critical values on opposite sides

36 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Requirement for Application The population MUST be normally distributed to hold (even when using large samples) This requirement is very strict!

37 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 1.When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data. 2.When the original set of data is unknown and only the summary statistics (n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample standard deviation. Round-Off Rules for Confidence Intervals Used to Estimate  or  2

38 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example Direct Computation: Chi-Squared Calculator (df = 39) Suppose the scores a test follow a normal distribution. Given a sample of size 40 with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation.

39 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Stat → Variance → One Sample → with Summary Example Suppose the scores a test follow a normal distribution. Given a sample of size 40 with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation.

40 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Enter parameters, then click Next Be sure to enter the sample variance s 2 (not s) Sample Variance Example Suppose the scores a test follow a normal distribution. Given a sample of size 40 with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation.

41 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Select Confidence Interval, enter Confidence Level, then click Calculate Example Suppose the scores a test follow a normal distribution. Given a sample of size 40 with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation.

42 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Remember: The result is the C.I for the Variance σ 2 Take the square root for Standard Deviation σ Variance Upper Limit: UL σ 2 Variance Lower Limit: LL σ 2 CI = ( LL σ 2, UL σ 2 ) = (16.2, 39.9) CI = ( LL σ 2, UL σ 2 ) = (4.03, 6.32) σ σ2σ2 Example Suppose the scores a test follow a normal distribution. Given a sample of size 40 with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation.

43 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Determining Sample Sizes The procedure for finding the sample size necessary to estimate  2 is based on Table 7-2 You just read the required sample size from an appropriate line of the table.

44 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Table 7-2

45 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example We want to estimate the standard deviation . We want to be 95% confident that our estimate is within 20% of the true value of . Assume that the population is normally distributed. How large should the sample be? For  95% confident and within 20% From Table 7-2 (see next slide), we can see that 95% confidence and an error of 20% for  correspond to a sample of size 48. We should obtain a sample of 48 values.

46 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. For  95% confident and within 20%