17a+Ch11 - BIT 2405 Quantitative Methods I BIT 2405...

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Unformatted text preview: BIT 2405 Quantitative Methods I BIT 2405 Quantitative Methods I Chapter 11.1 Inferences About A Population Variance This Week This Week Lectures Hawkes Today Optional HW911 and REQUIRED Quiz 3 are Posted. Next Week and Beyond. Next Week and Beyond. Lectures Hawkes I suggest you suggest work on Hawkes EARLY EARLY 9.1 is not required but you may want but to do the instruct. to Email me if you want to take the Early Test 3. Questions? Questions? Chapter 9 & 11.1 Chapter 9 & 11.1 Hypothesis Tests This Week KEY Lecture # Chapter TEXT Quantitative Methods I, Anderson, Sweeney & Williams Hawkes Certifications due on Monday on @ 11:59pm ! Hawkes Learning Systems: Statistics Certifications Section Topic 11.1 9.2 Hypothesis Testing Proportions: P Value 9.3 Hypothesis Testing Proportions: z Value Hypothesis Testing Means: P Value Hypothesis Testing Means: z Value Hypothesis Testing Means: t Value Hypothesis Testing Inferences about a Population Variance Today 17a Ch11.ppt Section Topic 9.4 17b Hypothesi sTesting.p pt 9 & 11 Overview and Summary of All the Hypothesis Tests 9.5 9.6 9.9 Chapter 11 Chapter 11 Inferences About Population Variances 11.1 Inference about a Population Variance s Inferences about the Variances of Two Populations Inferences About a Population Variance Inferences About a Population Variance s s s Chi­Square Distribution Interval Estimation of σ Hypothesis Testing 2 Chi Chi is Pronounced is ‘ kī Chi­Square Distribution Chi­Square Distribution s The chi­square distribution is the sum of squared standardized normal random variables such as (z1)2+(z2)2+(z3)2 and so on. The chi­square distribution is based on sampling from a normal population. The sampling distribution of (n ­ 1)s2/σ has a chi­ square distribution whenever a simple random sample of size n is selected from a normal population. 2 We can use the chi­square distribution to develop interval estimates and conduct hypothesis tests about a population variance. Examples of Sampling Distribution of (n ­ 1)s /σ Examples of Sampling Distribution of ( 2 2 With 2 degrees of freedom With 5 degrees of freedom With 10 degrees of freedom 0 (n − 1) s 2 σ2 The Chi­Square Distribution Is NOT SYMMETRICAL nor Centered on ZERO! Examples of Sampling Distribution of (n ­ 1)s /σ Examples of Sampling Distribution of ( 2 2 With 2 degrees of freedom With 5 degrees of freedom With 10 degrees of freedom 0 (n − 1) s 2 σ2 When df > 100, X2 becomes approximately normally distributed (mean = n and variance = 2n). We will NOT have sample sizes > 101 so you can use the X2 tables. Chi­Square Distribution s s χa We will use the notation to denote the value for the chi­square distribution that provides an area of a to the 2 χa right of the stated value. 2 For example, there is a .95 probability of obtaining a χ 2 (chi­square) value such that 2 2 χ.975 ≤ χ 2 ≤ χ.025 0.975 - 0.025 = 0.950 Interval Estimation of σ 2 2 χ.975 ≤ ( n − 1)s 2 .025 .025 95% of the possible χ 2 values 0 2 χ.975 σ2 2 ≤ χ.025 2 χ.025 χ2 Looking up χ in the Table. Looking up 2 1. A rea given is the Upper Tail Area f or a χ 2 value. 1. CHI-SQUARE Distribution CHI-SQUARE i s NOT SYMMETRICAL . NOT Interval Estimation of σ Interval Estimation of 2 There is a (1 – α) probability of obtaining a χ 2 value such that 2 2 2 s χ (1−α / 2) ≤ χ ≤ χα / 2 s Substituting (n – 1)s2/σ 2 for the χ 2 we get 2 χ (1−α / 2) s (n − 1) s 2 2 ≤ ≤ χα / 2 σ2 Performing algebraic manipulation we get ( n − 1) s2 ( n − 1) s2 ≤ σ2 ≤ 2 2 χα /2 χ (1− α / 2) Pay close attention to which X2 value you put where! Pay Interval Estimation of σ Interval Estimation of s 2 Interval Estimate of a Population Variance ( n − 1) s2 ( n − 1) s 2 ≤ σ2 ≤ 2 2 χα /2 χ (1− α / 2) where the χ 2 values are based on a chi­square values are based on a chi­square distribution with n ­ 1 degrees of freedom and where 1 ­ α is the confidence coefficient. Interval Estimation of σ Interval Estimation of s Interval Estimate of a Population Standard Deviation Taking the square root of the upper and lower limits of the variance interval provides the confidence interval for the population standard deviation. (n − 1) s 2 (n − 1) s 2 ≤σ ≤ 2 2 χα /2 χ (1−α / 2) Interval Estimation of σ Interval Estimation of s 2 Example: Buyer’s Digest (A) Buyer’s Digest rates thermostats manufactured for home temperature control. In a recent test, 10 thermostats manufactured by ThermoRite were selected and placed in a test room that was maintained at a temperature of 68oF. The temperature readings of the ten thermostats are shown on the next slide. Interval Estimation of σ Interval Estimation of 2 s Example: Buyer’s Digest (A) We will use the 10 readings below to develop a 95% confidence interval estimate of the population variance. Thermostat 1 2 3 4 5 6 7 8 9 10 Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2 Interval Estimation of σ 2 χ 2 ≤ χ 2 ≤ χ.025 2 .975 For n ­ 1 = 10 ­ 1 = 9 d.f. and α = .05 Selected Values from the Chi-Square Distribution Table Area in Upper Tail D egrees of Freedom 5 6 7 8 9 10 .99 0.554 0.872 1.239 1.647 2.088 .975 0.831 1.237 1.690 2.180 2.700 .95 1.145 1.635 2.167 2.733 3.325 .90 1.610 2.204 2.833 3.490 4.168 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 The The .10 9.236 10.645 12.017 13.362 14.684 2 χ.975 value .05 11.070 12.592 14.067 15.507 16.919 .025 12.832 14.449 16.013 17.535 19.023 .01 15.086 16.812 18.475 20.090 21.666 Interval Estimation of σ Interval Estimation of 2 χ 2 ≤ χ 2 ≤ χ.025 2 .975 For n ­ 1 = 10 ­ 1 = 9 d.f. and α = .05 2.700 ≤ χ ≤ χ 2 .025 0 2.700 Area in Upper Tail = .975 2 .025 χ2 Interval Estimation of σ 2 χ 2 ≤ χ 2 ≤ χ.025 2 .975 For n ­ 1 = 10 ­ 1 = 9 d.f. and α = .05 Selected Values from the Chi-Square Distribution Table Area in Upper Tail D egrees of Freedom 5 6 7 8 9 10 .99 0.554 0.872 1.239 1.647 2.088 .975 0.831 1.237 1.690 2.180 2.700 .95 1.145 1.635 2.167 2.733 3.325 .90 1.610 2.204 2.833 3.490 4.168 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 The The .10 9.236 10.645 12.017 13.362 14.684 2 χ.025 value .05 11.070 12.592 14.067 15.507 16.919 .025 12.832 14.449 16.013 17.535 19.023 .01 15.086 16.812 18.475 20.090 21.666 Interval Estimation of σ 2 χ 2 ≤ χ 2 ≤ χ.025 2 .975 n ­ 1 = 10 ­ 1 = 9 degrees of freedom and α = .05 2.700 ≤ χ 2 ≤ 19.023 .025 0 2.700 Area in Upper Tail = .025 19.023 χ2 Interval Estimation of σ Interval Estimation of s 2 Sample variance s2 provides a point estimate of σ 2. ∑ ( xi − x ) 2 6. 3 s= = = . 70 n −1 9 2 s A 95% confidence interval for the population variance is given by: (10 − 1). 70 (10 − 1). 70 2 ≤σ ≤ 19. 02 2. 70 2 .33 < σ 2 < 2.33 Formula ( n − 1) s2 ( n − 1) s2 ≤ σ2 ≤ 2 2 χα /2 χ (1− α / 2) 2.700 ≤ χ 2 ≤ 19.023 Hypothesis Testing Hypothesis Testing About a Population Variance s Lower­Tail Test •Hypotheses 2 H 0 : σ 2 ≥ σ 0 2 H a : σ 2 < σ 0 2 σ0 where is the hypothesized value for the population variance •Test Statistic (n − 1) s 2 2 χ obs = σ 02 Hypothesis Testing About a Population Variance s Lower­Tail Test (continued) •Rejection Rule Critical value approach: 2 2 Reject H0 if χ ≤ χ (1−α ) obs p­Value approach: Reject H0 if p­value < α 2 χ(1−α ) where is based on a chi­square distribution with n ­ 1 d.f. Hypothesis Testing About a Population Variance s Upper­Tail Test •Hypotheses H0 : σ 2 ≤ σ 2 0 Ha : σ 2 > σ 2 0 2 σ0 where is the hypothesized value for the population variance •Test Statistic ( n − 1) s 2 χ2 = σ2 obs 0 Hypothesis Testing About a Population Variance s Upper­Tail Test (continued) •Rejection Rule Critical value approach: Reject H0 if χ 2 ≥ χ α2 obs p­Value approach: Reject H0 if p­value < α 2 χα where is based on a chi­square distribution with n ­ 1 d.f. Hypothesis Testing About a Population Variance s Two­Tail Test •Hypotheses H0 : σ 2 = σ 2 0 Ha : σ 2 ≠ σ 2 0 2 σ0 where is the hypothesized value for the population variance •Test Statistic ( n − 1) s 2 χ2 = σ2 obs 0 Hypothesis Testing About a Population Variance s Two­Tail Test (continued) •Rejection Rule Critical value approach: 2 2 Reject H0 if χ 2 ≤ χ (1−α / 2 ) or χ 2 ≥ χα / 2 obs obs p­Value approach: Reject H0 if p­value < α 2 χ (1−α / 2) and χ α2 / 2 where are based on a chi­square distribution with n ­ 1 d.f. Hypothesis Testing About a Population Variance s Example: Buyer’s Digest (B) Recall that Buyer’s Digest is rating ThermoRite thermostats. Buyer’s Digest gives an “acceptable” rating to a thermostat with a temperature variance of 0.5 or less. We will conduct a hypothesis test (with α = .10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”. Hypothesis Testing Hypothesis Testing About a Population Variance s Example: Buyer’s Digest (B) Using the 10 readings, we will conduct a hypothesis test (with α = .10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”. Thermostat 1 2 3 4 5 6 7 8 9 10 Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2 Hypothesis Testing About a Population Variance s Hypotheses H 0 : σ 2 ≤ 0.5 H a : σ 2 > 0.5 s Rejection Rule Reject H0 if χ 2 > 14.684 obs Upper­ tail test Hypothesis Testing About a Population Variance For n ­ 1 = 10 ­ 1 = 9 d.f. and α = .10 Selected Values from the Chi-Square Distribution Table Area in Upper Tail D egrees of Freedom 5 6 7 8 9 10 .99 0.554 0.872 1.239 1.647 2.088 .975 0.831 1.237 1.690 2.180 2.700 .95 1.145 1.635 2.167 2.733 3.325 .90 1.610 2.204 2.833 3.490 4.168 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 The The 2 χ.10value .10 9.236 10.645 12.017 13.362 14.684 .05 11.070 12.592 14.067 15.507 16.919 .025 12.832 14.449 16.013 17.535 19.023 .01 15.086 16.812 18.475 20.090 21.666 Hypothesis Testing About a Population Variance s Rejection Region χ2 = obs ( n − 1)s 2 σ2 9s 2 = .5 Area in Upper Tail = .10 0 14.684 χ2 Reject H0 Hypothesis Testing About a Population Variance s Test Statistic The sample variance s 2 = 0.7 9(.7) χ= = 12.6 .5 obs 2 s Conclusion Because χ 2 = 12.6 is less than 14.684, we cannot obs reject H0. The sample variance s2 = .7 is insufficient evidence to conclude that the temperature variance for ThermoRite thermostats is unacceptable. Hypothesis Testing About a Population Variance s Using the p­Value • The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate p­value is less than .90 (χ 2 = 4.168) and greater than .10 (χ 2 = 14.684). • Because the p –value > α = .10, we cannot reject the null hypothesis. • The sample variance of s 2 = .7 is insufficient evidence to conclude that the temperature variance is unacceptable (>.5). Hypothesis Testing About a Population Variance s 9(.7) Using the p­Value 2 χ obs = = 12.6 • The rejection region for the ThermoRite .5 thermostat example is in the upper tail; thus, the appropriate p­value is less than .90 (χ 2 = 4.168) and greater than .10 (χ 2 = 14.684). • Because the p –value > α = .10, we cannot reject the null hypothesis. • The sample variance of s 2 = .7 is insufficient evidence to conclude that the temperature variance is unacceptable (>.5). Hypothesis Testing About a Population Variance s Using the p­Value • The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate p­value is less than .90 (χ 2 = 4.168) and greater than .10 (χ 2 = 14.684). • Because the p –value > α = .10, we cannot reject the null hypothesis. • The sample variance of s 2 = .7 is insufficient evidence to conclude that the temperature variance is unacceptable (>.5). Recap of Hypothesis Testing Recap of Hypothesis Testing ...
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