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The percent of persons (ages five and older) in each state who speak a language at home other than English is approximately exponentially distributed with a mean of 9.848. Suppose we randomly pick a state.

a. Define the random variable.X=_________________________________.

b. Is Xcontinuous or discrete?

c.X~________

d.μ=________

e.σ=________

f. Draw a graph of the probability distribution. Label the axes.

g. Find the probability that the percent is less than 12.

h. Find the probability that the percent is between eight and 14.

i. The percent of all individuals living in the United States who speak a language at home other than English is 13.8.

i. Why is this number different from 9.848%?

ii. What would make this number higher than 9.848%?

Short Answer

Expert verified

a. The percent of persons (ages five and older) in each state who speak a language at home other than English

b. the random variable Xis exponentially distribution, hence the Xwill be continuous.

c. X~1.07

d.μ=9.848

e. localid="1651433623625" σ=9.848

f. The graph is drawn

g. P(X<12)=0.7042

h. P(14<x<8)=0.2025

i.i. This should be because of the higher value of standard deviation which is present in the collected data.
ii. The given percentage is higher than 9.848because there are huge chances that each state will have English speakers higher and lower than the mean percentage.

Step by step solution

01

introduction

The percent of persons (ages five and older) in each state who speak a language at home other than English is approximately exponentially distributed with a mean of9.848.

02

explanation (part a)

The random variableXan be defined as the per cent of persons (ages five and older) in each state who speak a language at home other than English.

03

Explanation (part b)

Continous as the random variable Xis exponentially distributed.

04

Explanation (part c)

The random variable Xis exponentially distributed and can be defined as below:

=X~Exp19.848=X~Expm=X~Exp0.1015

05

Explanation (part d)

The mean can be represented asμ=9.848

06

Explanation (part e)

The standard deviation can be calculated as-

σ=μ=9.848

07

Explanation (part f)

Given that X~Exp(0.1015)som=0.1015
And, the general form of the probability density function of the exponential distribution is given below,

f(x)=memxf(x)=0.1015e0.1015x

The maximum value of f(x)which will lie on the y-axis and at x=0will be:

f(x)=(0.1015)e-(0.1015)(0)=0.1015

The value of f(x)for different values of x, we get

x
f(x)
-40.06763
-30.074855
-20.082852
-10.091703
00.1015
10.112343
20.124345
30.137629
40.152331

From the above table, the graph of the probability distribution is given as:

08

Explanation (part g)

It is known that the cumulative distribution function of the exponential distribution is,

p(X<x)=1-e-mx

The probability less than the per cent is less than 12is,

localid="1651432803210" p(X<12)=1-e-(0.1015×12)p(X<12)=1-0.2958p(X<12)=0.7042

09

Explanation (part h)

The required probability can be calculated as,

P(14<x<8)=P(X<14)-P(X<8)P(14<x<8)=(1-e-(0.1015×14))-(1-e-(0.1015×8))P(14<x<8)=0.4440-0.2415P(14<x<8)=0.2025

10

Explanation (part i)

i. This is due to the bigger value of standard deviation which is present in the collected data.
ii. Due to the huge chance that each state will have English speakers higher and lower than the mean percentage the given percentage is higher than9.848

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