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Lecture 11

01:960:401 Lecture 11: 00104-Basic Stats for Research-2016-07-05
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Department
Statistics
Course
01:960:401
Professor
Micheal Miniere
Semester
Summer

Description
Chapter 14 Sections 1, 2, 3: ANOVA Recall in Chapter 10 we tested the hypotheses In this chapter we test the hypothesis HI: There is at least one difference among the pairs of population means. Assumptions we make: 1. The populations have approximately normal distributions. 2. The samples are random samples. 3. The samples are independent. 4. The standard deviations from the populations are equal. (it has been shown that as long as the sample sizes are nearly equal, the variances can differ by amounts that make the largest up to nine times the smallest, and ANOVA still is reliable. Now suppose we have three groups of data Grand Mean Over All ni Total scores #of groups It turns out that the total variation about the grand mean is embodied in the sum of sum of squared deviations, ier X-ry called the total sum of squares SST so SST Xr- or equivalently, it can be shown algebraically that SST The computations ofANOVA involve partitioning the total variation about the grand mean into two components: 1. The variation that originates from the differences among the group means, i.e., Yu, Yn, Tc, etc., called (SSG) (the group sum of squares). 2. The random variation within the groups themselves called error, SSE) (error sum of squares) We do this by partitioning SST into the group sum of squares, SSG, and the error sum of squares, SSE. Thus we have SST SSG +SSE Now the formula for SSG is as follows: SSG Chapter 14 Sections 1, 2, 3: ANOVA Recall in Chapter 10 we tested the hypotheses In this chapter we test the hypothesis HI: There is at least one difference among the pairs of population means. Assumptions we make: 1. The populations have approximately normal distributions. 2. The samples are random samples. 3. The samples are independent. 4. The standard deviations from the populations are equal. (it has been shown that as long as the sample sizes are nearly equal, the variances can differ by amounts that make the largest up to nine times the smallest, and ANOVA still is reliable. Now suppose we have three groups of data Grand Mean Over All ni Total scores #of groups It turns out that the total variation about the grand mean is embodied in the sum of sum of squared deviations, ier X-ry called the total sum of squares SST so SST Xr- or equivalently, it can be shown algebraically that SST The computations ofANOVA involve partitioning the total variation about the grand mean into two components: 1. The variation that originates from the differences among the group means, i.e., Yu, Yn, Tc, etc., called (SSG) (the group sum of squares). 2. The random variation within the groups themselves called error, SSE) (error sum of squares) We do this by partitioning SST into the group sum of squares, SSG, and the error sum of squares, SSE. Thus we have SST SSG +SSE Now the formula for SSG is as follows: SSGconditions with the following results. The letters, A, M, and L denote automatic, manual, and lockup torque. At the 0.05 significance level, test the claim that the mean fuel consumption values are the same 23 23 29 26 20 25 24 23 24 We will do this in clas Example: A pilot does extensive bad weather flying and decides to buy a battery-powered radio as an independent backup for his regular radios, which depend on the airplanes electrical system. Ha has the choice of 3 brands that vary in cost. His obtains the sample data shown. He cts 5 batt d test recharging is necessary. Do the 3 brands have the same mean usable time before recharging is required? Test at the 0.05 significance level pud lsts in L Brand X Brand Y 26.0 29.0 30.0 28.5 26.3 27.3 27.6 29.2 d :0.05 25.9 27.1 29.8 28.2 27.0 We will do this in clas 0,24 0,05 ent sampl SSE do not SST SSG SSE Now the degrees of freedom for SST i DFT nr. -1 The degrees of freedom for k groups (SSG) is DFG k So the degrees of freedom for in-group variation (SSE) is DFE DFT DFG which is DFE Now we find the variance estimate (mean squares) for groups and error. S
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