Question

mat painter
font
ali
total
h i j k l m n o

summary
groups count sums average variance
northeast 56 108 1.92857 1.12208
south 99 166 1.67677 0.9965
midwest 81 174 2.14815 1.40278
west 56 138 2.46429 2.03506

anova
source of vari ss df ms f p-value f crit
between region
within region
total

Explanation:

Step1: Calculate Total Count

Sum the counts of all groups: \( 56 + 99 + 81 + 56 \).
\( 56 + 99 = 155 \), \( 155 + 81 = 236 \), \( 236 + 56 = 292 \).

Step2: Calculate Total Sums

Sum the sums of all groups: \( 108 + 166 + 174 + 138 \).
\( 108 + 166 = 274 \), \( 274 + 174 = 448 \), \( 448 + 138 = 586 \).

Step3: Calculate Total Average (Optional, but for completeness)

Total average = \( \frac{\text{Total Sums}}{\text{Total Count}} = \frac{586}{292} \approx 2.00685 \).

Step4: Calculate Between - Groups Degrees of Freedom (df)

Number of groups (\( k \)) = 4. Between - groups df = \( k - 1 = 4 - 1 = 3 \).

Step5: Calculate Within - Groups Degrees of Freedom (df)

Within - groups df = \( \text{Total Count}-k = 292 - 4 = 288 \).

Step6: Calculate Total Degrees of Freedom (df)

Total df = Between - groups df+Within - groups df = \( 3 + 288 = 291 \).

Step7: Calculate Between - Groups Sum of Squares (SS)

First, find the grand mean (\( \bar{X}_{GM} \)): \( \bar{X}_{GM}=\frac{586}{292}\approx2.00685 \).
Then, \( SS_{between}=\sum_{i = 1}^{k}n_i(\bar{X}_i-\bar{X}_{GM})^2 \)
For NorthEast: \( n_1 = 56 \), \( \bar{X}_1 = 1.92857 \), \( (\bar{X}_1-\bar{X}_{GM})^2=(1.92857 - 2.00685)^2\approx(- 0.07828)^2\approx0.00613 \), \( n_1(\bar{X}_1-\bar{X}_{GM})^2 = 56\times0.00613\approx0.3433 \)
For South: \( n_2 = 99 \), \( \bar{X}_2 = 1.67677 \), \( (\bar{X}_2-\bar{X}_{GM})^2=(1.67677 - 2.00685)^2\approx(-0.33008)^2\approx0.10895 \), \( n_2(\bar{X}_2-\bar{X}_{GM})^2 = 99\times0.10895\approx10.7861 \)
For Midwest: \( n_3 = 81 \), \( \bar{X}_3 = 2.14815 \), \( (\bar{X}_3-\bar{X}_{GM})^2=(2.14815 - 2.00685)^2\approx(0.1413)^2\approx0.01996 \), \( n_3(\bar{X}_3-\bar{X}_{GM})^2 = 81\times0.01996\approx1.6168 \)
For West: \( n_4 = 56 \), \( \bar{X}_4 = 2.46429 \), \( (\bar{X}_4-\bar{X}_{GM})^2=(2.46429 - 2.00685)^2\approx(0.45744)^2\approx0.20925 \), \( n_4(\bar{X}_4-\bar{X}_{GM})^2 = 56\times0.20925\approx11.718 \)
Sum these up: \( SS_{between}=0.3433 + 10.7861+1.6168 + 11.718\approx24.4642 \)

Step8: Calculate Within - Groups Sum of Squares (SS)

\( SS_{within}=\sum_{i = 1}^{k}(n_i - 1)s_i^2 \)
For NorthEast: \( (n_1 - 1)s_1^2=(56 - 1)\times1.12208 = 55\times1.12208\approx61.7144 \)
For South: \( (n_2 - 1)s_2^2=(99 - 1)\times0.9965 = 98\times0.9965\approx97.657 \)
For Midwest: \( (n_3 - 1)s_3^2=(81 - 1)\times1.40278 = 80\times1.40278\approx112.2224 \)
For West: \( (n_4 - 1)s_4^2=(56 - 1)\times2.03506 = 55\times2.03506\approx111.9283 \)
Sum these up: \( SS_{within}=61.7144+97.657 + 112.2224+111.9283\approx383.5221 \)

Step9: Calculate Total Sum of Squares (SS)

\( SS_{total}=SS_{between}+SS_{within}=24.4642 + 383.5221\approx407.9863 \)

Step10: Calculate Mean Square Between (MS)

\( MS_{between}=\frac{SS_{between}}{df_{between}}=\frac{24.4642}{3}\approx8.1547 \)

Step11: Calculate Mean Square Within (MS)

\( MS_{within}=\frac{SS_{within}}{df_{within}}=\frac{383.5221}{288}\approx1.3317 \)

Step12: Calculate F - statistic

\( F=\frac{MS_{between}}{MS_{within}}=\frac{8.1547}{1.3317}\approx6.123 \)

Step13: Calculate P - Value (using F - distribution with df1 = 3, df2 = 288)

Using an F - distribution table or calculator, the P - value for \( F = 6.123 \), \( df1 = 3 \), \( df2 = 288 \) is approximately 0.0005 (this can be calculated more precisely with software like Excel's FDIST function: \( FDIST(6.123,3,288)\approx0.0005 \))

Step14: Calculate F - critical value (for a common alpha = 0.05)

Using an F - distribution table or calculator, for \( \alpha = 0.05 \), \( df1 = 3 \), \( df2 = 288 \), the F - critical value is approximately 2.64 (can be found using Excel's FINV function: \( FINV(0.05,3,288)\approx2.64 \))

Answer:

The completed ANOVA table (approximate values) is:

Source of Variation	SS	df	MS	F	P - Value	F crit
Within Region	383.52	288	1.33
Total	407.99	291

(Note: Values are approximate due to rounding during calculations. More precise values can be obtained using software for ANOVA calculations.)