2. sum of squares suppose you are interested ...
2. sum of squares suppose you are interested in the relationship between a persons age and the amount of community activism in which he or she is involved. you randomly select nine people from each of three different age categories and ask each person to estimate the number of hours spent engaged in any type of community activism or volunteer activity during the past 30 days (x1). you obtain the following data summary from your investigation: age group 18 - 30 31 - 50 over 50 ∑x1 = 51 ∑x1 = 53 ∑x1 = 57 ∑x1² = 303 ∑x1² = 313 ∑x1² = 373 in the following questions you will practice using the formulas from the text to calculate the main components of an anova. according to the summary data table, the total sample size is n = _ and the overall mean is x = _. the three category means are x1 = _, x2 = _, and x3 = _. calculate the sum of squares total (sst), the sum of squares between (ssb), and the sum of squares within (ssw) this data set. sst = _. ssb = _. ssw = _.
Answer
# Explanation:
## Step1: Calculate total sample size N
There are 9 people in each of 3 age - groups, so $N=9\times3 = 27$.
## Step2: Calculate the overall mean $\bar{X}$
The sum of all $X_i$ values is $\sum_{i = 1}^{3}\sum X_{i}=51 + 53+57=161$. Then $\bar{X}=\frac{161}{27}\approx5.963$.
## Step3: Calculate the category means
For age - group 18 - 30: $\bar{X}_1=\frac{\sum X_{1}}{9}=\frac{51}{9}\approx5.667$.
For age - group 31 - 50: $\bar{X}_2=\frac{\sum X_{2}}{9}=\frac{53}{9}\approx5.889$.
For age - group over 50: $\bar{X}_3=\frac{\sum X_{3}}{9}=\frac{57}{9}\approx6.333$.
## Step4: Calculate SST
$SST=\sum_{i = 1}^{3}\sum X_{i}^{2}-\frac{(\sum_{i = 1}^{3}\sum X_{i})^{2}}{N}=303 + 313+373-\frac{161^{2}}{27}=989-\frac{25921}{27}\approx989 - 959.9 = 29.1$.
## Step5: Calculate SSB
$SSB=\sum_{j = 1}^{3}n_j(\bar{X}_j-\bar{X})^2$, where $n_j = 9$ for $j = 1,2,3$.
$SSB=9\times[(5.667 - 5.963)^2+(5.889 - 5.963)^2+(6.333 - 5.963)^2]$
$=9\times[(- 0.296)^2+(-0.074)^2+(0.37)^2]$
$=9\times(0.087616 + 0.005476+0.1369)$
$=9\times0.229992\approx2.07$.
## Step6: Calculate SSW
$SSW=SST - SSB=29.1-2.07 = 27.03$.
# Answer:
$N = 27$, $\bar{X}\approx5.963$, $\bar{X}_1\approx5.667$, $\bar{X}_2\approx5.889$, $\bar{X}_3\approx6.333$, $SST\approx29.1$, $SSB\approx2.07$, $SSW\approx27.03$