Question

the distribution of the tuition, fees, and room and board charges of a random sample of public 4 - year degree - granting postsecondary institutions is shown in the pie chart. then use the table to estimate the sample mean and the sample standard deviation of the data. use $26249.50 as the midpoint for $25,000 or more. complete the frequency distribution for the data. (type integers or decimals. do not round.) class $15,000 - $17,499 $17,500 - $19,999 $20,000 - $22,499 $22,500 - $24,999 $25,000 or more x 16249.5 18749.5 21249.5 23749.5 26249.5 f 7 11 13 9 6 the sample mean is x = $193495.4 (round to the nearest cent.)

Explanation:

Step1: Calculate mid - points

The mid - points for the classes are given as $x = [16249.5,18749.5,21249.5,23749.5,26249.5]$.

Step2: Calculate frequencies

The frequencies are $f=[7,11,13,9,6]$.

Step3: Calculate the product of mid - points and frequencies

Let $xf$ be the product of $x$ and $f$. We calculate $xf$ for each pair:
For the first pair: $16249.5\times7 = 113746.5$
For the second pair: $18749.5\times11=206244.5$
For the third pair: $21249.5\times13 = 276243.5$
For the fourth pair: $23749.5\times9=213745.5$
For the fifth pair: $26249.5\times6 = 157497$

Step4: Calculate the sum of frequencies

$\sum f=7 + 11+13+9+6=46$

Step5: Calculate the sum of $xf$

$\sum xf=113746.5+206244.5+276243.5+213745.5+157497=967477$

Step6: Calculate the sample mean

The sample mean $\bar{x}=\frac{\sum xf}{\sum f}=\frac{967477}{46}\approx21032.11$

Step7: Calculate the squared differences

Let $d=x-\bar{x}$. Then $d^2=(x - \bar{x})^2$.
For $x = 16249.5$, $d_1=16249.5 - 21032.11=-4782.61$, $d_1^2 = 22873384.71$
For $x = 18749.5$, $d_2=18749.5 - 21032.11=-2282.61$, $d_2^2=5210254.31$
For $x = 21249.5$, $d_3=21249.5 - 21032.11 = 217.39$, $d_3^2 = 47258.41$
For $x = 23749.5$, $d_4=23749.5 - 21032.11=2717.39$, $d_4^2 = 7384274.31$
For $x = 26249.5$, $d_5=26249.5 - 21032.11 = 5217.39$, $d_5^2=27220144.71$

Step8: Calculate the product of $d^2$ and $f$

For the first pair: $22873384.71\times7 = 160113692.97$
For the second pair: $5210254.31\times11 = 57312797.41$
For the third pair: $47258.41\times13=614359.33$
For the fourth pair: $7384274.31\times9 = 66458468.79$
For the fifth pair: $27220144.71\times6=163320868.26$

Step9: Calculate the sum of $d^2f$

$\sum d^2f=160113692.97+57312797.41+614359.33+66458468.79+163320868.26=447810286.76$

Step10: Calculate the sample variance

The sample variance $s^2=\frac{\sum d^2f}{n - 1}=\frac{447810286.76}{46-1}=\frac{447810286.76}{45}\approx9951339.71$

Step11: Calculate the sample standard deviation

The sample standard deviation $s=\sqrt{s^2}=\sqrt{9951339.71}\approx3154.57$

Answer:

The sample mean is approximately $\$21032.11$ and the sample standard deviation is approximately $\$3154.57$