Suppose the length of newborn babies, measured from heel to crown, is normally d

Question

Suppose the length of newborn babies, measured from heel to crown, is normally distributed with a mean of 20 inches and a standard deviation of 2 inches. A hospital researcher collects a random sample of 66 newborn babies' lengths.

a) What does the Central Limit Theorem tell us about the distribution of the average of these 66 lengths?

b) What is the probability that the sample average of these 66 lengths is less than 19.6 or greater than 20.7?

c) What is the 33rd percentile of this sample average?

d) Suppose the researcher would like to collect a sample such that the average of their lengths has standard deviation=0.15. How many newborn babies must be sampled in order to achieve this?

Explanation / Answer

a) as per Central Limit Theorem  distribution of the average of these 66 lengths will be normal. with mean=20

and std error =std deviaiton/(n)1/2 =0.246

b) P(X<19.6)+P(X>20.7)=1-P(19.6<X<20.7)=1-P((19.6-20)/0.246<Z<(20.7-20)/0.246)

=1-P(-1.6248<Z<2.8434)=1-(0.9978-0.0521)=0.0543

c) for 33rd percentile; z=-0.4399

hence corresponding value =20-0.4399*0.246=19.8917

d)sample size =(std deviation/std error)2 =(2/0.15)2 =~178

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