Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Using the graph of the cumulative distributive function below, find the:

(a) mean

(b) median

(c) mode

(d) midrange

(e) third quartile for the random variable

CDF

I really just want help with parts (b) and (e).

I started this question by creating a table from the graph and then back calculating the probability function:

$$ \begin{array}{l|l} \hline X & 3 & 4 & 6 & 6.5 & 7 & 7.5 \\ \hline F(x) & 0 & 0.2 & 0.4 & 0.8 & 0.9 & 1 \\ \hline Pr(X) & 0 & 0.2 & 0.2 & 0.4 & 0.1 & 0.1 \\ \hline \end{array} $$

(a) $\mu_X=(4)(0.2) + 6(0.2) + 6.5(0.4) + 7(0.1) = 7.5(0.1) = 6.05$

(b) I'm confused on how to calculate the median from the table. I know the median is the center value of a data set or the 50th percentile ($Q_2$).

To calculate percentiles:

If $x_1, x_2, \cdots, x_n$ are $n$ data points arranged in ascending order, then $x_i$ corresponds to the $\left(100 \cdot \frac{i}{n+1}\right)^{th}$ percentile.

So this is what I did: $Q_2 = 50 = 100 \cdot \frac{i}{11}\tag{1}$

I set the $n=10$ because I the probabilities are all in tenths, so I thought you could imagine there are $10$ possible outcomes. So you could say there are two $4$s and $6$s, four $6.5$'s, and one $7$ and one $7.5$.

Since there is an even number of data points (namely $10$), the median will be the mean of the $5$th and $6$th data point. Since $6.5$ is the $5$th, $6$th, $7$th, and $8$th data point, the mean is $6.5$. Is this best or correct approach?

(c) mode = $6.5$

(d) midrange = $(4+7.5)/2 = 5.75$

(e) Not getting this one.

$Q_3: 75 = 100 \cdot \frac{i}{11}$

Solving, $i=8.25$. This is just the index. Now, we have to interpolate to get the value for the $8.25$ index:

$\cfrac{8.25 - 8}{x-6.5} = \cfrac{9-8}{7-6.5}$

$\implies x=6.625$ which is wrong according to the book..... This should be $6.5$. I checked the errata but nothing there. Am I getting the wrong answer? Also , is there a faster way of doing this?

To do this method you have to manually count and find which value of $X$ belongs to the index $8$ and $9$ which can be time consuming if you have a table with $36$ data points. Is there a faster way?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.