Question

The weights of 2-year old children follow a normal distribution with an average weight of 26 lbs and a standard deviation of 1.5 lbs.

You select a random sample of 100 2-year old children.

What is the shape of the sampling distribution for the average of 100 2-year olds?

What is the mean of the sampling distribution for the average of 100 2-year olds?

What is the standard deviation of the sampling distribution for the average of 100 2-year olds?

Explain the 68-95-99.7 rule in this case (i.e. in what range you of values do you expect the average weight of 100 2-year olds to fall 68 %, 95%, 99.7% of the time (i.e. in many samples of size 100)? Show your calculations.

What is the probability that the average weight of 100 randomly selected 2-year olds is less than 26.04 lbs?

What is the probability that the average weight of 100 randomly selected 2-year olds is more than 26.04 lbs?

Answer 1

Population:

Mean, M = 26

Standard Deviation, S = 1.5

Sample:

Size, n = 100

Let us answer the questions now-

What is the shape of the sampling distribution for the average of 100 2-year olds?

As the sample size, n > 30, this will follow a normal distribution

What is the mean of the sampling distribution for the average of 100 2-year olds?

Sample Mean, m = Population Mean, M = 26

What is the standard deviation of the sampling distribution for the average of 100 2-year olds?

Sample Standard Deviation, s = S/sqrt(n) = 1.5/sqrt(100) = 0.15

Explain the 68-95-99.7 rule in this case (i.e. in what range you of values do you expect the average weight of 100 2-year olds to fall 68 %, 95%, 99.7% of the time (i.e. in many samples of size 100)? Show your calculations.

In the context of the sample,

68% of the 2-year olds have weights between m-s and m+s (26-0.15 and 26+0.15) i.e. 25.85 and 26.15

95% of the 2-year olds have weights between m-2s and m+2s (26-2*0.15 and 26+2*0.15) i.e. 25.7 and 26.3

99.7% of the 2-year olds have weights between m-3*s and m+3*s (26-3*0.15 and 26+3*0.15) i.e. 25.55 and 26.45

What is the probability that the average weight of 100 randomly selected 2-year olds is less than 26.04 lbs?

P(X<26.04)

Z-score = (26.04-26)/0.15 = 0.266

P(Z<0.266) = 0.6045

What is the probability that the average weight of 100 randomly selected 2-year olds is more than 26.04 lbs?

P(X>26.04)

Z-score = (26.04-26)/0.15 = 0.266

P(Z>0.266) = 1-P(Z<0.266) = 1-0.6045 = 0.3955

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