Does average self-reported weekly income differ between male and female full-time workers in Sydney?

401077 Introduction to Biostatistics, autumn 2019
Statistics
Student Name:
Instructor Name:
Course Number:
27 April 2019
.

Question 1 (12 marks)
Research question: Does average self-reported weekly income differ between male and
female full-time workers in Sydney?
a) Using R Commander, graph the distributions of income for males and females
separately. Write a sentence describing the shape of these distributions. Repeat for the
logarithm transformed values of income (variable: log_income). (2 marks)
Answer
This chart presents histogram for the income levels for the male and the females. As
can visualized, it is clear that the distribution of income for both the females and the
males is skewed (right skewed to be specific).
Next chart presents histogram for the logarithm of the income levels (log_income) for
the male and the females. As can visualized, it is clear that the distribution of income
for both the females and the males unlike the previous ones is close to normal
distribution (the two plots have bell-shaped curves).

b) The research question could be addressed by conducting either an independent
samples t-test on the difference in mean ‘income’ between genders or an independent
samples t-test on the difference in mean ‘log_income’ between genders. Which of
these two alternatives would you chose? Justify your choice. (2 marks)
Answer
The income levels were found to be heavily skewed hence I would go for an
independent samples t-test on the difference in mean ‘log_income’ between genders
which is as shown by the histogram is close to a normal distribution. Independent
samples t-test is a parametric test and it requires data to be normally distributed.
c) Using R Commander, implement the analysis you chose in b). Present your results
using the 5-step method. (4 marks)
Answer
Step 1: Stating the hypothesis
H0 : μm=μf
H A : μm ≠ μf
Where μm =average logincome for the males
μf =average log income for the fe males

Step 2: Checking for the assumptions
i) All the observations were found to be independent of one another
ii) The dependent variable is measured on a ratio scale
iii) The dependent variable was found to be normally distributed
iv) Homogeneity of variances (Equal variances); we found out that there was
homogeneity of variances which is one of the required assumptions.
Step 3: Calculating the test statistic
An independent samples t-test was computed and
the results presented in the table below;
Step 4: Evaluating the results
The above output shows that the observed
difference in the means is not statistically
significant t(496) = 0.046, p = .963. We fail to
reject the null hypothesis (H0) in favour of the
alternative hypothesis (HA).
Step 5: Interpreting the results
The results indicate that the income levels for the
male and female employees are the same and that there is no evidence of variation in
the salaries of the employees based on gender.
d) The logarithm transformation changes the numeric values of the data but does not
change the order of data values. Therefore, a non-parametric hypothesis test (which
ranks the data) will give the same answer whether conducted on ‘income’ or ‘log-
> leveneTest(log_income
~ sex, data = workhours)
Levene's Test for
Homogeneity of Variance
(center = median)
Df F value Pr(>F)
group 1 0.3712 0.5426
496
> t.test(log_income~sex)
Welch Two
Sample t-test
data: log_income by sex
t = 0.0464, df = 464.732,
p-value = 0.963
alternative hypothesis:
true difference in means
is not equal to 0
95 percent confidence
interval:
-0.1193908 0.1251654
sample estimates:
mean in group male
mean in group female
7.253744
7.250856