##   bia.di bii.di bit.di che.de che.di elb.di wri.di kne.di ank.di sho.gi che.gi wai.gi nav.gi hip.gi thi.gi bic.gi for.gi kne.gi cal.gi ank.gi wri.gi age  wgt   hgt sex
## 1   42.9   26.0   31.5   17.7   28.0   13.1   10.4   18.8   14.1  106.2   89.5   71.5   74.5   93.5   51.5   32.5   26.0   34.5   36.5   23.5   16.5  21 65.6 174.0   1
## 2   43.7   28.5   33.5   16.9   30.8   14.0   11.8   20.6   15.1  110.5   97.0   79.0   86.5   94.8   51.5   34.4   28.0   36.5   37.5   24.5   17.0  23 71.8 175.3   1
## 3   40.1   28.2   33.3   20.9   31.7   13.9   10.9   19.7   14.1  115.1   97.5   83.2   82.9   95.0   57.3   33.4   28.8   37.0   37.3   21.9   16.9  28 80.7 193.5   1

Male Height Histogram

Female Height Histogram

1) Both distributions appear to be normal shaped. They have a single mode and a peak near the mode. Most of the observations are close to the mode and there are very few at each end. It appears symmetric. The mean for men is 177.7453441, but the mean for the women is 164.8723077, meaning that men are taller, on average. Visually, it appeared that men have a more peaked distribution. Rescaling them evenly, like above, makes women appear to have a more peaked distribution(higher kurtosis). The coefficient of variation for women is 0.039695, but is 0.0404153 for men. They have a very similar shape.

fhgtmean <- mean(fdims$hgt)
fhgtsd   <- sd(fdims$hgt)
fwgtmean <- mean(fdims$wgt)
fwgtsd   <- sd(fdims$wgt)
hist(fdims$hgt, probability = TRUE)
x <- 140:190
y <- dnorm(x = x, mean = fhgtmean, sd = fhgtsd)
lines(x = x, y = y, col = "blue")

2) Based on the given histogram with the overlayed normal curve, our female height distribution appears to be normal.

Male Height QQ Plot

Female Height QQ Plot

Simulated Normal Plot with Same N as fdims

Simulated Q-Q Plot with Same N as fdims

3) Not all of the points of our simulated normal q-q plot fall on the line either. This is because the further you get to the edges of a normal curve, the more susceptible it is to being different than the predicted value. It looks very similar to the real data.

4) The plot for female heights looks rather similar to the collection of simulated plots. It seems likely that famale heights are nearly normal.

Female Weight QQ Plot

5) Female weights do not appear to be normal. The plot is skewed to the right and the q-q plot shows an increasing slope on the right. The skewness of female weights is 1.13517 and the kurtosis is quite high at 2.5473685.

6) a)What is the probability that a randomly chosen young adult female is taller than 5 feet 7 inches (about 170 cm)? b) What is the probability that a randomly chosen young adult female is heavier than 187.4 pounds (about 85kg)? Based on the thoeoretical normal probability, 21.6666932% will be above 170cm, and 22.3076923% will be above 170cm based on the empirical distribution. Based on the thoeoretical normal probability, 0.5582733% will be above 85 kg, and 1.5384615% will be above 85 kg based on the empirical distribution.

While height pretty close to the theoretical distribution at the chosen level, weight is 3 times more probable at the chosen level under the empirical distribution as it is under the theoretical distribution.

On your own:

Match the histograms to q-q plots: B,C,D,A
Plots C and D have a slight stepwise pattern. This can often be induced by effects of bin width and would go away with more data or a different separation of intervals.
Female knee diameter is right skewed. It can be seen in the plot immediately below and in the data further below. (It’s measurement 18.)

We run a skewness test (3rd central moment/sd cubed) to find interesting distributions that fail being normal, because they are skewed. The furthest skewed results were: For women, measure 2 was left skewed and measure 12 was right skewed. For men, measure 2 was also left skewed and measure 22 was right skewed. The skewness measures are followed by graphs of the most skewed empirical distributions. 

##              [,1]         [,2]
##  [1,]  0.17837591 -0.213130152
##  [2,] -0.53652532 -0.160432916
##  [3,]  0.01632648 -0.074331970
##  [4,]  1.02352424  0.333781880
##  [5,]  0.75151304 -0.072128637
##  [6,] -0.11878093 -0.047814511
##  [7,]  0.13636070  0.302625197
##  [8,]  1.12369055  0.224233113
##  [9,] -0.24807130  0.008921324
## [10,]  0.82180456  0.079207286
## [11,]  0.75238426 -0.145941585
## [12,]  1.23194455  0.632111931
## [13,]  0.69115590  0.380340257
## [14,]  0.66413493  0.416230828
## [15,]  0.79459273  0.501533354
## [16,]  0.83186972 -0.039297202
## [17,]  0.82130206 -0.088354716
## [18,]  0.98648410  0.317081617
## [19,]  0.52460267  0.138630728
## [20,]  0.33311660  0.281379726
## [21,]  0.36165993  0.164301733
## [22,]  1.18532895  1.030518986
## [23,]  1.13517003  0.289412189
## [24,]  0.06536036  0.102900857
## [25,]          NA           NA

                 Female

Biiliac Diameter (pelvic breadth) in Centimeters      Waist Girth in Centimeters


                 Male

Biiliac Diameter (pelvic breadth) in Centimeters      Age

Female biiliac diameter has cv=0.0836602. Female biiliac diameter has kurtosis= 0.9390832.
Female waist girth has cv= 0.1087016. Female waist girth has kurtosis= 1.9312862.
Male biiliac diameter has cv=0.0735845. Male biiliac diameter has kurtosis=0.9090916.
Male age has cv= 0.3205584. Male age has kurtosis=0.3803568