Answer to Question #107139 in Statistics and Probability for tara

Question

An electric store has received a shipment of 20 table radios that have connections
for the iPhone. 12 of these have 2 slots and the others have a single slot. Suppose, 6
of the 12 radios are randomly selected to store. Let, X be the number of stored
radios with two slots.
a. What kind of s distribution does X have? (include all the values of the
parameters)
b. Compute, P(X ≤ 3 ).
c. Calculate the mean and standard deviation of X.
d. Calculate the probability that X exceeds its mean value by less than one
standard deviation.
e. Plot the Histogram of X and comment on the shape of the distribution.

Answer 1

a. X be the number of stored radios with two slots has hypergeometric distribution

$$P(X=x)=h(x;n, M, N)={\dbinom{M}{x}\dbinom{N-M}{n-x} \over \dbinom{N}{n}}$$

Given that $N=20, M=12, n=6.$

b.

$P(X\leq3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)$

$$P(X=0)={\dbinom{12}{0}\dbinom{20-12}{6-0} \over \dbinom{20}{6}}={1(28) \over 38760}={28 \over 38760}$$

$$P(X=1)={\dbinom{12}{1}\dbinom{20-12}{6-1} \over \dbinom{20}{6}}={12(56) \over 38760}={672 \over 38760}$$

$$P(X=2)={\dbinom{12}{2}\dbinom{20-12}{6-2} \over \dbinom{20}{6}}={66(70) \over 38760}={4620 \over 38760}$$

$$P(X=3)={\dbinom{12}{3}\dbinom{20-12}{6-3} \over \dbinom{20}{6}}={220(56) \over 38760}={12320\over 38760}$$

$$P(X\leq3)={28 \over 38760}+{672 \over 38760}+{4620 \over 38760}+{12320 \over 38760}=$$

$$={17640 \over 38760}={147\over 323}\approx0.4551$$

c. Calculate the mean and standard deviation of X. 

$$\mu=E(X)=n\cdot{M \over N}=6\cdot{12 \over 20}=3.6$$

$$Var(X)=\sigma^2=({N-n \over N-1})\cdot n\cdot{M \over N}(1-{M \over N})=$$

$$=({20-6\over 20-1})\cdot 6\cdot{12 \over 20}(1-{12 \over 20})={504 \over 475}$$

$$\sigma=\sqrt{{504 \over 475}}\approx1.03$$

d.

$$\mu+\sigma\approx3.6+1.03=4.63$$

$$P(3.6<X\leq4.63)=P(X=4)=$$

$$={\dbinom{12}{4}\dbinom{20-12}{6-4} \over \dbinom{20}{6}}={495(28) \over 38760}={13860\over 38760}=$$

$$={231\over 646}\approx0.3576$$

e.

Moderately negative skewed.