7) If you know that the average weight of the people in a population is 160 lbs,

Question

7) If you know that the average weight of the people in a population is 160 lbs, and the variance is 25, what is the probability that the average weight in a sample of 16 people is over 165 lbs? What is the probability that the average weight in a sample of 36 is over 162.5 lbs? What is the probability that the average weight will be between 158 and 161 lbs (n = 12)? (3)

a) >165lbs average (sample of 16):

b) >162.5lbs average (sample of 36):

c) P(158 < x < 161):

8) For the population in question 7), which is the lower bound for the interval that contains the highest 5% of values? The upper bound for the interval that contains the lowest 10% of values? (2)

Explanation / Answer

Let X be the random variable that weight of the people.

Given that X has mean = 160 lbs and variance = 25

sd = sqrt(variance) = sqrt(25) = 5

Here we have to use Central limit theorem.

It states that for large n the distribution of sample mean goes to normal with mean is population mean and standard deviation is sd/sqrt(n).

n = 16

mu = 160

sd = 5/sqrt(16) = 1.25

FOr n = 36

sd = 5 / sqrt(36) = 0.83

what is the probability that the average weight in a sample of 16 people is over 165 lbs?

Here we have to find P(Xbar > 165)

Now we have we to find z-score for Xbar = 165.

z-score is definde as,

z = (Xbar - mu) / (sd / sqrt(n))

z = (165 - 160) / 1.25 = 4

Now we have to find P(Z > 4)

This probability we can find in EXCEL.

syntax :

=1 - NORMSDIST(z)

where z is z-score.

P(Z >4) = 0.0000317

What is the probability that the average weight in a sample of 36 is over 162.5 lbs?

n = 36

Here we have to find P(Xbar > 162.5)

z-score for Xbar = 162.5 is,

z= (162.5 - 160) / 0.83 = 3

Now we have to find P(Z > 3)

P(Z >3) = 0.0013

What is the probability that the average weight will be between 158 and 161 lbs (n = 12)?

n = 12

Here we have to find P(158 < Xbar < 161)

z-scores for Xbar = 158 and 161 are,

z = (158 - 160) / 1.44 = -1.39

z = (161 - 160) / 1.44 = 0.69

Now we have to find P(-1.39 < Z < 0.69)

P(-1.39 < Z < 0.69)= P(Z < 0.69) - P(Z < -1.39)

This probability also we can find in EXCEL.

syntax :

=NORMSDIST(z)

where z is z-score

P(-1.39 < Z < 0.69) = 0.7558 - 0.0829 = 0.6729

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