Analysis of Film Thickness Variables

Question 1 (25 marks):
a) Describe the structure of the film.txt data. (2 marks)
Answer
It is a data frame with 160 observations and 5 variables. All the five variables (Number,
Top Right, Top Left, Bottom Right, and Bottom Left) are integer variables.
b) Produce and interpret univariate QQ plots and histograms and univariate ShapiroWilks
tests of normality for each of the four film thickness variables. Which is the most non-
normally distributed variable? (5 marks)
Answer
QQ Plots
Looking at the above plots, we can see that the QQ plot for the Top Right and the Bottom
Right are almost linear. This suggests that the Top Right and the Bottom Right are close
to being normally distributed. The other two plots (Top Left Plot and the Bottom Left
Plot) don’t seem to be linear implying that they are far from normal distribution.

Histograms
For the histogram plots above, we see that the histogram for the Top Right and the
Bottom Right are almost bell-shaped implying that the tow variables are close to normal
distribution. On the other hand, the histogram for the Top left and Bottom Left looks like
they are skewed to the right (longer tails to the right), this shows that the two (Top left
and Bottom Left are not normally distributed but are rather skewed to the right.
Shapiro-Wilk tests for normality
The table below presents the Shapiro-Wilk tests for the four variables (Top Right, Top
Left, Bottom Right and Bottom Left). As can be seen, the p-value for the Top Right is
0.091 (a value greater than 5% level of significance), we therefore fail to reject the null
hypothesis and conclude that the variable Top Right is normally distributed. Also we can
see that the p-value for the Bottom Right is 0.291 091 (a value greater than 5% level of
significance), we therefore fail to reject the null hypothesis and conclude that the variable
Bottom Right is normally distributed.

On the other hand, the p-value for Top left and Bottom left are less than 0.05, we
therefore reject the null hypothesis and conclude that the variable Top left and Bottom
Left are not normally distributed.
c) Produce and
thickness variables. What is an inherent problem with
using these plots to assess MVN? (3 marks)
Answer
> shapiro.test(TopRight)
Shapiro-Wilk
normality test
data: TopRight
W = 0.9854, p-value =
0.09075
> shapiro.test(TopLeft)
Shapiro-Wilk
normality test
data: TopLeft
W = 0.9733, p-value =
0.003413
>
shapiro.test(BottomRight
)
Shapiro-Wilk
normality test
data: BottomRight
W = 0.9896, p-value =
0.2905
>
shapiro.test(BottomLeft)
Shapiro-Wilk
normality test
data: BottomLeft
W = 0.9678, p-value =
0.0008824