2.1.1 estimate the mean wage and mean wage of Married/Notmarried professionals in the U.S.

We can get the mean estimation of wage. And after dividing the data into married and not married, it is easy to estimate the mean of married/notmarried. From the value we get, it is easy to find the married professionals have high wage.

##  Two-Sided 95% Confidence Interval: Arithmetic Mean
## 
##     n nNA pNA    M   SD  Low  Upp
##   526   0  0% 5.90 3.69 5.58 6.21

##  Two-Sided 95% Confidence Interval: Arithmetic Mean
## 
##     n nNA pNA    M   SD  Low  Upp
##   320   0  0% 6.57 3.99 6.13 7.01

##  Two-Sided 95% Confidence Interval: Arithmetic Mean
## 
##     n nNA pNA    M   SD  Low  Upp
##   206   0  0% 4.84 2.89 4.45 5.24

2.1.2 estimate the difference in mean wage of Married/Nomarried professionls

Now we estimate the difference to make the thought above more clear. Calculate the confidence interval of difference of mean of Married/Notmarried people, and use plot to show the details. It is obvious that the married have higher wage.

##  Two-Sided 95% Confidence Interval: Difference in Means from Independent Samples
## 
##   Between   n nNA    M   SD M.Diff  Low  Upp
##    1      206   0 4.84 2.89                 
##    2      320   0 6.57 3.99   1.73 1.14 2.32

2.2.1 estimate the proportion of married professionals in different wage intervals.

After that, I want to research by another way. I divide the wage into 5 intervals, and estimate the proportion of married from different intervals. If the proportion increases while interval increases, it will show the thought from another side. Firstly, we can easily draw a proportion of the married information in different intervals, it looks like our thought is correct. Then we can estimate the confidence interval to research deeply. The confidence interval of last two interval we need use “wilson” method to instead of “wald”. Because wald method is a simple method to use the proportion of sample instead of population proportion. And it will let the upper higher to be 1.

##  Two-Sided 95% Confidence Interval: Proportion
## 
##     n nNA pNA  Prop   Low   Upp
##   526   0  0% 0.608 0.566 0.649

##  Two-Sided 95% Confidence Interval: Proportion
## 
##     n nNA pNA  Prop   Low   Upp
##   292   0  0% 0.517 0.460 0.574

##  Two-Sided 95% Confidence Interval: Proportion
## 
##     n nNA pNA  Prop   Low   Upp
##   182   0  0% 0.709 0.639 0.770

##  Two-Sided 95% Confidence Interval: Proportion
## 
##    n nNA pNA  Prop   Low   Upp
##   36   0  0% 0.722 0.576 0.869

##  Two-Sided 95% Confidence Interval: Proportion
## 
##    n nNA pNA  Prop   Low   Upp
##   11   0  0% 0.909 0.623 0.984

##  Two-Sided 95% Confidence Interval: Proportion
## 
##   n nNA pNA  Prop   Low   Upp
##   5   0  0% 0.800 0.376 0.964

I use ggplot() to draw the result of calculation. Although the proportion is truly in the trend of increasing, but the margin of error is too high for some intervals, because they are small samples.

tmp = data.frame(groups,proportion,marginError_Low,marginError_Upper)
tmp$groups <- factor(tmp$groups, levels = groups)
ggplot(tmp, aes(x = groups, y = proportion,color = groups)) + 
    geom_errorbar(aes(ymin = marginError_Low, ymax = marginError_Upper), width = 0.1) +
    geom_point(size = 4, shape = 17, col = "hotpink") +
    xlab("wage interval") +
    ylab("married proportion estimation") +
    ggtitle("Estimation of proportion of married in different intervals of wage")

2.2.2 estimate the difference in proportions of married professionals above/below mean of wage.

Now we can estimate the difference in proportion of married above/below mean of wage to simply the thought above and enlarge the sample in different intervals. From the four fold plot it can be simply find higher wage has more married professionals. Then after we calculate the confidence interval for difference of proportion. But the margin of error is still about 0.08 which is 0.08/0.21 = 38%.

data("wage1")
wage1$married <- sub("Notmarried", 0, wage1$married)
wage1$married <- sub("Married", 1, wage1$married)
wage1$married <- sapply(wage1$married, as.numeric)

above <- subset(wage1, wage > meanWage)
below <- subset(wage1, wage < meanWage)

ci.prop.diff(below$married, above$married)

##  Two-Sided 95% Confidence Interval: Difference in Proportions from Independent Samples
## 
##   Between   n nNA    p p.Diff  Low  Upp
##    1      325   0 0.53                 
##    2      201   0 0.74   0.21 0.12 0.29