Question 1001120
a) 


Info: "One class has space available for 30 students, eight classes have space for 60 students, one has space for 70 students, and four classes have space for 100 students. "


Compute the weighted mean based on the info

1*30 + 8*60 + 1*70 + 4*100 = 980

980/(1+8+1+4) = 980/14 = 70


Average number of students per class is 70 students. You have the correct answer.


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b)

X = number of students in a single class


<table  border=1><tr><th>X</th><th>P(X)</th></tr><tr><td>30</td><td>1/14</td></tr><tr><td>60</td><td>8/14 = 4/7</td></tr><tr><td>70</td><td>1/14</td></tr><tr><td>100</td><td>4/14 = 2/7</td></tr></table>

The PDF is simply the table shown above. It's the set of all possible probabilities for every scenario.
Take note how there are only 2 columns in this PDF. The input column X and the output column P(X)



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c)

Now we introduce the X*P(X) column


<table border=1><tr><th>X</th><th>P(X)</th><th>X*P(X)</th></tr><tr><td>30</td><td>1/14</td><td>2.142857</td></tr><tr><td>60</td><td>4/7</td><td>34.28571</td></tr><tr><td>70</td><td>1/14</td><td>5</td></tr><tr><td>100</td><td>2/7</td><td>28.571429</td></tr></table>

sum the values in the last column

2.142857+34.28571+5+28.571429 = 69.999996

Due to rounding error, we don't get exactly 70 if we use the decimal values. If you stick with fractions the whole time, then you'll get 70 exactly

Either way, the answer is 70 like in part (a)

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d)

from parts (a) and (c), we know the mean is mu = 70


Now add on the columns X - mu,  (X-mu)^2, and (X - mu)^2*P(X)


<table border=1><tr><th>X</th><th>P(X)</th><th>X*P(X)</th><th>X - mu</th><th>(X - mu)^2</th><th>(X - mu)^2*P(X)</th></tr><tr><td>30</td><td>1/14</td><td>2.142857143</td><td>-40</td><td>1600</td><td>114.2857143</td></tr><tr><td>60</td><td>4/7</td><td>34.28571429</td><td>-10</td><td>100</td><td>57.14285714</td></tr><tr><td>70</td><td>1/14</td><td>5</td><td>0</td><td>0</td><td>0</td></tr><tr><td>100</td><td>2/7</td><td>28.57142857</td><td>30</td><td>900</td><td>257.1428571</td></tr></table>


Add up the values in the last column:

114.2857143 + 57.14285714 + 0 + 257.1428571 = 428.57142854


The variance of X is approximately 428.57142854


Then take the square root of the variance to get sqrt(428.57142854) = 20.7019667795116



So the standard deviation of X is approximately 20.7019667795116




Here is a page that offers another example on finding the standard deviation:

<a href = "https://www.ltcconline.net/greenl/courses/201/probdist/random.htm">https://www.ltcconline.net/greenl/courses/201/probdist/random.htm</a>