Answer to Question #107988 in Statistics and Probability for Yusuf

Question

Boy Girl Total
Speech Therapist (ST) 90 30
Neuropsychologist (NP) 45 15
Psychiatrist (P) 30
Total 55

Question 29
Which one of the following statements is incorrect?
(1) P(ST and Boy) = 0.409
(2) P(P and Girl) = 0.045
(3) P(ST|Boy) = 0.545
(4) P(NP|Girl) = 0.068
(5) P(NP|Boy) = p(NP|Girl) = P(NP)

Question 30 Which one of the following statements is incorrect? (1) The events Neuropsychologist and Boy are independent. (2) The events Neuropsychologist and Girl are independent. (3) The events Psychiatrist and Boy are dependant. (4) Autism Spectrum Disorder in more common in boys than in girls. (5) The events Speech Therapist and Psychiatrist are mutually exclusive.

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} & Boys & Girls & Total \\ \hline Speech\ Therapist \ (ST) & 90 & 30 & 120\\ \hline Neuropsychologist\ (NP) & 45 & 15 & 60\\\hline Psychiatrist\ (P) & 30 & 10 & 40\\ \hdashline Total & 165 & 55 & 220 \end{array}$$

Question 29

(1)

$$P(ST \ and\ B) =P(ST\cap B)={90\over 220}={9\over 22}\approx0.409$$

(2)

$$P(P \ and\ G) =P(ST\cap B)={10\over 220}={1\over 22}\approx0.045$$

(3)

$$P(ST | B) ={90\over 165}={6\over 11}\approx0.545$$

(4)

$$P(NP| G) ={15\over 55}={3\over 11}\approx0.273$$

(5)

$$P(NP| B) ={45\over 165}={3\over 11}=P(NP| G)$$

$$P(NP) ={60\over 220}={3\over 11}=P(NP| B)=P(NP| G)$$

The statement '(4) P(NP|Girl) = 0.068' is incorrect.

Question 30

(1)

$$P(B)={165\over 220}={3\over 4}, P(NP)={60\over 220}={3\over 11}$$

$$P(NP\cap B)={45\over 220}={9\over 44}$$

$$P(B)\cdot P(NP)={3\over 4}\cdot {3\over 11}={9\over 44}=P(NP\cap B)$$

The events Neuropsychologist and Boy are independent.

(2)

$$P(G)={55\over 220}={1\over 4}, P(NP)={60\over 220}={3\over 11}$$

$$P(NP\cap G)={15\over 220}={3\over 44}$$

$$P(G)\cdot P(NP)={1\over 4}\cdot {3\over 11}={3\over 44}=P(NP\cap G)$$

The events Neuropsychologist and Girl are independent.

(3)

$$P(B)={165\over 220}={3\over 4}, P(P)={40\over 220}={2\over 11}$$

$$P(P\cap B)={30\over 220}={3\over 22}$$

$$P(B)\cdot P(P)={3\over 4}\cdot {2\over 11}={3\over 22}=P(P\cap B)$$

The events Psychiatrist and Boy are independant.

(4)

$$P(B)={3\over 4}>{1\over 4}=P(G)$$

Autism Spectrum Disorder in more common in boys than in girls.

(5)

$$ST\cap P=\empty$$

The events Speech Therapist and Psychiatrist are mutually exclusive.

The statement '(3) The events Psychiatrist and Boy are dependent' is incorrect