Answer to Question #120238 in Statistics and Probability for Josh Porter

Question

1. Suppose IQ scores are normally distributed with mean 100 and standard deviation 10. Which of the following is false?

A. Roughly 68% of people have IQ scores between 90 and 110.
B. A normal probability plot of IQ scores of a random sample of 1,000 people should show a straight line.

C. An IQ score greater than 130 is highly unlikely, but not impossible.

D. An IQ score of 80 is more unusual than an IQ score of 120.


2. A 2005 survey found that 7% of teenagers (ages 13 to 17) suffer from an extreme fear of spiders (arachnophobia). At a summer camp there are 10 teenagers sleeping in each tent. Assume that these 10 teenagers are independent of each other. What is the probability that at least one of them suffers from arachnophobia?

A. 72%
B. 82%
C. 42%
D. 62%
E. 52%

Answer 1

Let $X=$ IQ score: $X\sim N(\mu, \sigma^2).$ Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0,1).$

Given $\mu=100, \sigma=10.$

A.

$$P( 90<X<110)=P(\mu-\sigma<X<\mu+\sigma)\approx0.68$$

The statement A is true.

B. The statement B is false.

C.

$$P(X>130)=1-P(X\leq130)=$$

$$=1-P(Z\leq{130-100\over 10})=1-P(Z\leq3)\approx0.001350$$

The statement C is true.

D.

The probability $P(X=80)$ is undefined. The probability $P(X=120).$

The statement D is false.

Which of the following is false? 

The statement B and the statement D are false.

2. Let $X=$ the number of teenagers who suffers from arachnophobia: $X\sim Bin(n, p)$

$$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$$

Given $n=10, p=0.07$

$$P(X\geq1)=1-P(X=0)=$$

$$=1-\binom{10}{0}(0.07)^0(1-0.07)^{10-0}=$$

$$=1-(0.93)^{10}\approx0.5160(\approx\ 52\%)$$

E. 52%