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Best Point Estimate Section 7.4 Estimation of a Population Mean (s is unknown) _ The sample mean x is still the best point estimate of the population mean m. This section presents methods for estimating a population mean when the population standard deviation s is not known. 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Student t Distribution ( t-dist ) 2 Definition 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.” When σ is unknown, we must use the Student t distribution instead of the normal distribution. In this section: df = n – 1 Requires new parameter df = Degrees of Freedom 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. 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 4 Important Properties of the Student t Distribution 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. 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 α (area) P(t < -1) (Area under curve) 4. S.D. 0 -1 0 σ > 1 (Note: σ varies with df) 5. As df gets larger, the t-dist approaches the z-dist tα (Critical value) NOTE: The values depend on df Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 6 Student t Distributions for n = 3 and n = 12 z-Distribution and t-Distribution df = 2 Wider Spread df = 100 Almost the same As df increases, the t-dist approaches the z-dist 7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Progression of t-dist with df df = 2 df = 3 Choosing the Appropriate Distribution df = 4 s known and normally Use the normal (Z) distribution df = 6 df = 20 df = 5 df = 7 df = 50 df = 8 distributed population or s known and n > 30 s not known and normally Use t distribution distributed population or s not known and n > 30 Methods of Ch. 7 do not apply Population is not normally distributed and n ≤ 30 df = 100 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Calculating values from t-dist Calculating values from t-dist Stat → Calculators → T Enter Degrees of Freedom (DF) and t-score Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 10 12 Calculating values from t-dist P(t<-1) = 0.1646 Calculating values from t-dist when df = 20 tα = 1.697 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Margin of Error E for Estimate of m (σ unknown) Formula 7-6 𝑬 = 𝒕𝜶 2 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. C.I. for the Estimate of μ (With σ Not Known) 𝒔 𝒏 where t/2 has n – 1 degrees of freedom. 𝑬 = 𝒕𝜶 t/2 = The t-value separating the right tail so it has an area of /2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. when α = 0.05 df = 20 15 Finding the Point Estimate and E from a C.I. Point estimate of µ: 2 𝒔 𝒏 𝒅𝒇 = 𝒏 − 𝟏 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 16 Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s Note: Same parameters as example used in Section 7-3 7-3: Etimating a population mean: σ known Margin of Error: Using σ = 10 ( instead of s = 10.0 ) we found the 90% confidence interval: C.I. = (35.9, 40.9) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 18 Example: Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Direct Computation: Using StatCrunch 𝒅𝒇 = 𝒏 − 1 = 41 𝒕𝜶 2 = 𝒕0.05 = 1.6839 𝑬 = 𝒕𝜶 2 𝒔 10.0 = 1.6839 = 2.5983 𝒏 42 𝒙 − 𝑬 = 38.4 − 2.5983 = 35.8017 T Calculator (df = 41) 𝒙 + 𝑬 = 38.4 + 2.5983 = 40.9983 𝑪. 𝑰. = 𝒙 − 𝑬, 𝒙 + 𝑬 = (35.8, 41.0) Stat → T statistics → One Sample → with Summary 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example: Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Using StatCrunch Using StatCrunch Select Confidence Interval and enter Confidence Level, then click Calculate Enter Parameters, click Next 21 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Example: Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s Using StatCrunch Results Standard Error If σ known Used σ = 10 to obtain 90% CI: If σ unknown Used s = 10.0 to obtain 90% CI: Lower Limit (35.9, 40.9) Upper Limit (35.8, 41.0) From the output, we find the Confidence interval is CI = (35.8, 41.0) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notice: σ known yields a smaller CI (i.e. less uncertainty) 23 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 24 Best Point Estimate of s2 Section 7.5 Estimation of a Population Variance The sample variance s2 is the best point estimate of the population variance s2 This section presents methods for estimating a population variance s2 and standard deviation s. 25 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Best Point Estimate of s 26 The 2 Distribution ( 2-dist ) Pronounced “Chi-squared” The sample standard deviation s is the best point estimate of the population standard deviation s Also dependent on the number degrees of freedom df. 27 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 28 Calculating values from 2-dist Properties of the 2 Distribution 1. The chi-square distribution is not symmetric, unlike the z-dist and t-dist. Stat → Calculators → Chi-Squared 2. The values can be zero or positive, they are nonnegative. 3. Dependent on the Degrees of Freedom: df = n – 1 Chi-Square Distribution Chi-Square Distribution for df = 10 and df = 20 Use StatCrunch to Calculate values (similar to z-dist and t-dist) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 29 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 30 Example: Find the 90% left and right critical values (2L and 2R) of the 2-dist when df = 20 Calculating values from 2-dist Enter Degrees of Freedom DF and parameters ( same procedure as with t-dist ) Need to calculate values when the left/right areas are 0.05 ( i.e. α/2 ) 2L = 10.851 P(2 < 10)= 0.5595 when df = 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 31 Important Note!! 2R = 31.410 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 32 Confidence Interval for Estimating a Population Variance The 2-distribution is used for calculating the Confidence Interval of the Variance σ2 Take the square-root of the values to get the Confidence Interval of the Standard Deviation σ Note: Left and Right Critical values on opposite sides ( This is why we call it 2 instead of ) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 33 Confidence Interval for Estimating a Population Standard Deviation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 34 Requirement for Application The population MUST be normally distributed to hold (even when using large samples) This requirement is very strict! Note: Left and Right Critical values on opposite sides Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 35 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 36 Example Round-Off Rules for Confidence Intervals Used to Estimate s or s 2 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. 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. Direct Computation: 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. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 37 Example Chi-Squared Calculator (df = 39) 38 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 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. Using StatCrunch 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. Using StatCrunch Sample Variance Enter parameters, then click Next Be sure to enter the sample variance s2 (not s) Stat → Variance → One Sample → with Summary Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 39 Example 40 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 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. Using StatCrunch 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. Using StatCrunch Remember: The result is the C.I for the Variance σ2 Take the square root for Standard Deviation σ Variance Lower Limit: LLσ 2 Variance Upper Limit: ULσ σ2 CI = ( LLσ , ULσ ) = (16.2, 39.9) 2 Select Confidence Interval, enter Confidence Level, then click Calculate Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2 σ 41 2 CI = ( LLσ , ULσ ) = (4.03, 6.32) 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2 42 Table 7-2 Determining Sample Sizes The procedure for finding the sample size necessary to estimate s2 is based on Table 7-2 You just read the required sample size from an appropriate line of the table. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 43 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 44 For s 95% confident and within 20% Example We want to estimate the standard deviation s. We want to be 95% confident that our estimate is within 20% of the true value of s. Assume that the population is normally distributed. How large should the sample be? For s 95% confident and within 20% From Table 7-2 (see next slide), we can see that 95% confidence and an error of 20% for s correspond to a sample of size 48. We should obtain a sample of 48 values. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 45 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 46