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Chapter 11: Q14E (page 450)

Use the fact that \(E\left( {{X_{ij}}} \right) = \mu \pm {\alpha _i} + {\beta _j}\)with \(\Sigma {\alpha _i} = \Sigma {\beta _j} = 0\) to show that\(E\left( {{{\bar X}_{i \times }} - {{\bar X}_{..}}} \right) = {\alpha _i}\). , so that \({\hat \alpha _i} = {\bar X_{i \times }} - \bar X\)is an unbiased estimator for\({\alpha _i}\).

Short Answer

Expert verified

The solution of the given equation is unbiased estimator.

Step by step solution

01

Drive the equation using mean and variance

Let where the following holds

and where the are independent normally distributed random variable with mean 0 and variance\({\sigma ^2}\).

From this, the following is true

\(E\left( {{X_{ij}}} \right) = \mu + {\alpha _i} + {\beta _j},\)

which is also given in the exercise.

Denote with\({\bar X_i}\), the average of measurements obtained when factor A is held at level i

with \({\bar X_{ \cdot j}}\)the average of measurements obtained when factor B is held at level j

Observed values are denoted with small \(x\) instead of big X. The notations without line over X are just the sums.

02

Prove that the given equation is unbiased estimator

The following holds

It has been proved that

\({\hat \alpha _i} = {\bar X_{i.}} - {\bar X_{..}}\)

In unbiased estimator of\({\alpha _i}\).

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Most popular questions from this chapter

The power curves of Figures 10.5 and 10.6 can be used to obtain \(\beta = P\) (type II error) for the F test in two-factor ANOVA. For fixed values of\({\alpha _1},{\alpha _2}, \ldots ,{\alpha _1}\), the quantity \({f^2} = (J/I)\Sigma \alpha _i^2/{\sigma ^2}\) is computed. Then the figure corresponding to \({v_1} = I - 1\)is entered on the horizontal axis at the value\(\phi \), the power is read on the vertical axis from the curve labeled \({\nu _2} = (I - 1)(J - 1),\;and\;\beta = 1 - \)power.

a. For the corrosion experiment described in Exercise 2, find\(\beta \;when\;{\alpha _1} = 4,{\alpha _2} = 0,{\alpha _3} = {\alpha _4} = - 2\) and \(\sigma = 4\).Repeat for \({\alpha _1} = 6,{\alpha _2} = 0,{\alpha _3} = {\alpha _4} = - 3\) and\(\sigma = 4\)

b. By symmetry, what is \(\beta \)for the test of \({H_{0B}}\;versus\;{H_{aB}}\)in Example 11.1 when \({\beta _1} = .3,{\beta _2} = {\beta _3} = {\beta _4} = - .1\) and \(\sigma = .3\)?

In an experiment to see whether the amount of coverage of light-blue interior latex paint depends either on the brand of paint or on the brand of roller used, one gallon of each of four brands of paint was applied using each of three brands of roller, resulting in the following data (number of square feet covered)


  1. Construct the ANOVA table. (Hint: The computations can be expedited by subtracting 400 (or any other convenient number) from each observation. This does not affect the final results.)
  2. State and test hypotheses appropriate for deciding whether paint brand has any effect on coverage. Use=.05.
  3. Repeat part (b) for brand of roller.
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The water absorption of two types of mortar used to repair damaged cement was discussed in the article 鈥淧olymer Mortar Composite Matrices for Maintenance-Free, Highly Durable Ferrocement鈥� (J. of Ferrocement, 1984: 337鈥�345). Specimens of ordinary cement mortar (OCM) and polymer cement mortar (PCM) were submerged for varying lengths of time (5, 9, 24, or 48 hours) and water absorption (% by weight) was recorded. With mortar type as factor A (with two levels) and submersion period as factor B (with four levels), three observations were made for each factor level combination. Data included in the article was used to compute the sums of squares, which were SSA 5 322.667, SSB 535.623, SSAB 5 8.557, and SST 5 372.113. Use this information to construct an ANOVA table. Test the appropriate hypotheses at a .05 significance level.

An investigation of the machinability of beryllium-copper alloy using two different dielectric mediums and four different working currents resulted in the following data on material removal rate (this is a subset of the data that appeared in the article 鈥淪tatistical Analysis and Optimization Study on the Machinability of Beryllium Copper Alloy in Electro Discharge Machining,鈥� J. of Engr. Manufacture, 2012: 1847鈥�1861).

a. After constructing an ANOVA table, test at level .05 both the hypothesis of no medium effect against the appropriate alternative and the hypothesis of no working current effect against the appropriate alternative.

b. Use Tukey鈥檚 procedure to investigate differences in expected material removal rate due to different working currents (Q.05,4,3 = 6.825).

The article 鈥淭he Responsiveness of Food Sales to Shelf Space Requirements鈥� (J. Marketing Research, 1964: 63鈥�67) reports the use of a Latin square design to investigate the effect of shelf space on food sales. The experiment was carried out over a 6-week period using six different stores, resulting in the following data on sales of powdered coffee cream (with shelf space index in parentheses):

\({X_{ij(k)}} = \mu + {\alpha _i} + {\beta _j} + {\partial _k} + {\`o _{ij(k)}},\quad i,j,k = 1,2, \ldots ,N\)

Construct the ANOVA table, and state and test at level .01 the hypothesis that shelf space does not affect sales against the appropriate alternative.
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