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76. The attention span of a two-year-old is exponentially distributed with a mean of about eight minutes. Suppose we randomly survey 60 two-year-olds.

a. In words, X=

b. X~

c. In wordsX-=

d. X-~

e. Before doing any calculations, which do you think will be higher? Explain why.

i. The probability that an individual attention span is less than ten minutes.

ii. The probability that the average attention span for the 60 children in less than ten minutes?

f. Calculate the probabilities in part e.

g. Explain why the distribution for X- is not exponential.

Short Answer

Expert verified

a. X is the attention span of a two-year-old

b. X~Exp18

c. X-is the mean of the average attention span of two-year-old

d. X¯~N8,860

e. The probability that the average attention span for the60children in less than ten minutes will be higher

f. the probabilities in part e are 0.713and0.973

g. It follows a natural distribution

Step by step solution

01

Given Information

The mean is 8minutes

The standard deviation is 8

sample size n =60

02

Explanation Part (a)

X is defined as the attention span of a two-year-old

03

Explanation Part (b)

We know,

mean μx=8which is exponentially distributed

now using,

X~Exp1μx

On putting the values we get

X~Exp18

The distribution isX~Exp18

04

Explanation Part (c)

X-is defined as the mean of the average attention span of two-year-old

05

Explanation Part (d)

We know,

mean μx=8which is exponentially distributed

The standard deviation σx=8

sample size n =60

now using,

X¯~Nμx,σxnas it is normally distributed

Putting the values, we get the distribution

=X¯~N8,860

06

Explanation Part (e)

The standard deviation is more modest, so there is more region under the normal curve. For example 60when contrasted with an example of one (a solitary individual), the standard deviation will be less and the qualities will bunch all the more firmly around the mean, and larger qualities will be more uncommon.

Hence The probability that the average attention span for the 60children in less than ten minutes will be higher.

07

Explanation Part (f)

Considering that P(x<10)be the probability that the individual attention span is less than ten minutes.

Now for an exponential distribution with parameter 𝜆the probability isP(xa)=eaλ

localid="1652452836040" =P(x<10)=e100.125=0.713

The probability that an individual attention span is less than ten minutes is 0.713.

P(X-<10)is the probability that the average attention span for the 60 children is less than ten minutes.

Using a calculator we get,

localid="1652453012848" P(X-<10)=normalcd(10,8,1.033)=0.973

08

Explanation Part (g)

According to the central limit theorem, the distribution for X-is not exponential as the mean follows a normal distribution as n gets bigger.

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