Q1)

A/D) Analyze Data!

Consider for the following data, we would like to investigate two factors : feed-rate & depth-of-cut, each with 3 replicates; to understand surface-finish (Dependent Var).

We guess the population model is something like this :

\[ S_{f_{i}} = \mu + F_{r_{i}} + D_{c_{j}} + FD_{ij}+\epsilon_{ij} \\ \text{For n-levels: } i\in[1,3]; j\in[1,4] \\ \text{Def.} \\ S_{f_{i}} = \text{surface-finish Effect} \\ F_{r_{j}} = \text{feed-rate Effect} \\ D_{c_{i}} = \text{depth-of-cut Effect} \\ FD_{i} = \text{Interaction Effect} \\ \epsilon_{i} = \text{Rand. Noise} \\ \]

##                   Df Sum Sq Mean Sq F value   Pr(>F)    
## FeedRate           1 2970.4  2970.4   85.93 1.40e-10 ***
## DepthCut           1 2042.3  2042.3   59.08 9.22e-09 ***
## FeedRate:DepthCut  1  413.2   413.2   11.96  0.00156 ** 
## Residuals         32 1106.1    34.6                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Consider each effect :

FeedRate           1 2970.4  2970.4   85.93 1.40e-10 ***
  • We notice that FeedRate is the most statistically highly significant factor
DepthCut           1 2042.3  2042.3   59.08 9.22e-09 ***
  • We notice that DepthCut is the second most statistically highly significant factor
FeedRate:DepthCut  1  413.2   413.2   11.96  0.00156 **
  • The interaction effect is the least statistically highly significant factor

Overall we find that each of the factors & their interaction are significant. Additionally, we notice our estimates are rather precise – that is, \(MS_{error}\) is rather small.

B) Diagnostics & Interaction Plot:

Diagnostics :

Normality Ass.

What we notice, from the plot is that it is damn near perfect! This like as good of normally distributed data as we can ask for. So, the normality assumption is satisfied.

Constant Variance Ass.

I would say, this data appears to have approx. const. variance.

Model Adequacy :

We notice a parabolic nature to our data’s residuals, whereby \(\hat{S}_{f_{i}}\) underpredicts then overpredicts then underpredicts.


Interaction Plot : XXXXX

C) Point-Est. at Each feeding Rate :

## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
## # A tibble: 3 × 2
##   FeedRate `mean(SurfaceFinish)`
##      <dbl>                 <dbl>
## 1     0.2                   81.6
## 2     0.25                  97.6
## 3     0.3                  104.

Q2) CI-95% \(\mathbb{E}(\Delta x_i)\)

Well, recall he whole point of chapter 2 : compare two things – in other words, consider the most appropriate measures will be using T-Test.

Suppose we est. each variance is roughly equivalent so we can use \(S_p\)

Consider the equation :

\[ \Delta x_i \pm t_{\frac{\alpha}{2}, df} S_p\sqrt{\frac{1}{n_1} + \frac{1}{n _2}}\\ df = n_1+n_2-2 \\ \]

## [1] -16

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