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Chapter 4: Q 2E (page 264)

Suppose that 20 percent of the students who took acertain test were from schoolAand that the arithmetic average of their scores on the test was 80. Suppose alsothat 30 percent of the students were from school B and that the arithmetic average of their scores was 76. Suppose, finally, that the other 50 percent of the students were from schoolCand that the arithmetic average of their scores was 84. If a student is selected at random from the entiregroup that took the test, what is the expected value of herscore?

Short Answer

Expert verified

The expected score of a student is 80.8

Step by step solution

01

Given Information

Suppose that 20 percent of the students who took a certain test were from school Aand that the arithmetic average of their scores on the test was 80.

\(\)\(\begin{align}\Pr \left( A \right) &= 0.20\\E\left( {Score|A} \right) &= 80\end{align}\)

Suppose also that 30 percent of the students were from school B and that the arithmetic average of their scores was 76

\(\begin{align}\Pr \left( B \right) &= 0.30\\E\left( {Score|B} \right) &= 76\end{align}\)

Suppose, finally, that the other 50 percent of the students were from school Cand that the arithmetic average of their scores was 84

\(\begin{align}{l}\Pr \left( C \right) &= 0.50\\E\left( {Score|C} \right) &= 84\end{align}\)

02

Finding the expected score of the selected student

If X denotes the score of the selected student, then

\(\begin{align}E\left( X \right) &= E\left( {E\left( {X|School} \right)} \right)\\ &= \Pr \left( A \right)E\left( {X|A} \right) + \Pr \left( B \right)E\left( {X|B} \right) + \Pr \left( C \right)E\left( {X|C} \right)\\ &= 0.20 \times 80 + 0.30 \times 76 + 0.50 \times 84\\ &= 80.8\end{align}\)

The expected score of the selected student is 80.8

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Most popular questions from this chapter

Suppose that an observed value of X is equally likely to come from a continuous distribution for which the pdf is for from one for which the pdf is g. Suppose that \(f\left( x \right) > 0\) for \(0 < x < 1\) and \(f\left( x \right) = 0\) otherwise, and suppose also that \(g\left( x \right) > 0\) for \(2 < x < 4\) and \(g\left( x \right) = 0\) otherwise. Determine:

  1. the mean and
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Suppose that a particle starts at the origin of the whole line and moves along the line in jumps of one unit. For each jump, the probability is p (0 鈮� p 鈮� 1) that the particle will jump one unit to the left, and the probability is 1 鈭� p that the particle will jump one unit to the right. Find the expected value of the position of the particle after n jumps

Suppose that a point\({{\bf{X}}_{\bf{1}}}\)is chosen from the uniform distribution on the interval\(\left( {{\bf{0,1}}} \right)\)and that after the value\({{\bf{X}}_{\bf{1}}}{\bf{ = }}{{\bf{x}}_{\bf{1}}}\)is observed, a point\({{\bf{X}}_{\bf{2}}}\)is chosen from a uniform distribution on the interval\(\left( {{{\bf{x}}_{\bf{1}}}{\bf{,1}}} \right)\). Suppose further that additional variables\({{\bf{X}}_{\bf{3}}}{\bf{,}}{{\bf{X}}_{\bf{4}}}{\bf{,}}...\)are generated in the same way. Generally,\({\bf{j = 1,2,}}...{\bf{,}}\)after the value\({{\bf{X}}_{\bf{j}}}{\bf{ = }}{{\bf{x}}_{\bf{j}}}\)has been observed,\({{\bf{X}}_{{\bf{j + 1}}}}\)is chosen from a uniform distribution on the interval\(\left( {{{\bf{x}}_{\bf{j}}}{\bf{,1}}} \right)\). Find the value of\({\bf{E}}\left( {{{\bf{X}}_{\bf{n}}}} \right)\).

Consider the box of red and blue balls in Examples 4.2.4 and 4.2.5. Suppose that we sample n>1 balls with replacement, and let X be the number of red balls in the sample. Then we sample n balls without replacement, and we let Y be the number of red balls in the sample. Prove that\({\bf{{\rm P}}}\left( {{\bf{X = n}}} \right){\bf{ > {\rm P}}}\left( {{\bf{Y = n}}} \right)\)

Prove that the \(\frac{1}{2}\) quantile defined in Definition 3.3.2 is a median as defined in Definition 4.5.1.
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