Question

I need help with Part e. I have already constructed the histogram using R and Rstudio (picture below). I am just confused about the empirical rule and locating the data points on the intervals. If anyone could assist me, I'd appreciate it greatly! Thank you!

e radon concentration (in pCi/liter) data obtained from 40 houses in a 1.5.10. Th certain area are given below. 2.9 0.6 13.5 17.1 2.8 3.8 16.0 2.1 6.4 17.2 7.9 0.5 13.7 115 2.9 3.6 6.1 15.9 8.89.811.5 12.3 3.7 6.2 6.9 12.8 13.7 2.7 3.5 9.4 11.7 6.0 2.2 7.9 3.5 8.3 15.9 5.1 8.8 8.913.0 for any outliers. (c) Construct a box plot. (d) Construct a histogram and interpret. (e) Locate on your histogram x±s, 2s, and xH3s. Count the data points in each of the intervals x, x±s, x±2, and x±3s. How do these counts compare with the empirical rule? Histogram of radon CO 2寸 0 5 10 15 radon

Answer 1

From the histogram , we observe that the it is not symmetric , the distribution is positively skewed.

First find the mean $\bar{x}$ and sample standard deviation s of the given data set using the functions mean() and sd() in R respectively.

$\bar{x}$ = 8.34 and s = 4.92

Therefore

$\bar{x}$ - s = 8.34 - 4.92 = 3.42 and   $\bar{x}$ + s = 13.26

$\bar{x}$ - 2*s = 8.34 - 9.84 = -1.50 and $\bar{x}$ + 2*s = 8.34 + 9.84 = 18.18

$\bar{x}$ - 3*s = 8.34 - 14.76 = -6.42 and $\bar{x}$ + 3*s = 8.34 + 14.76 = 23.10

So locate -6.42,-1.50, 3.42, 13.26,18.18,23.10 on the histogram

In the range ( 3.42,13.26 ) there are 24 numbers in the data set are lies , so % of data numbers lies in interval (   $\bar{x}$ - s , $\bar{x}$ + s ) is 24/40 = 0.6 or 60%

According to Empirical rule approximately 68% of the observations fall within (   $\bar{x}$ - s , $\bar{x}$ + s )

In the range ( -1.50,18.18 ) there are all 40 numbers in the data set are lies , so % of data numbers lies in interval (   $\bar{x}$ -2*s , $\bar{x}$ +2* s ) is 40/40 = 1 or 100%

According to approximately 95% of the observations fall within   ( $\bar{x}$ -2 *s , $\bar{x}$ +2* s )

In the range ( -6.42,23.10 ) there are all 40 numbers in the data set are lies , so % of data numbers lies in interval (   $\bar{x}$ -3*s , $\bar{x}$ +3* s ) is 40/40 = 1 or 100%

According to approximately 99.7% of the observations fall within   ( $\bar{x}$ -3*s , $\bar{x}$ +3* s )

Empirical rules stated for the symmetric distribution, therefore from the above result we can say that the given distribution of the radon concentration is not symmetric.