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Practice Problems # 05

1. The Apgar score is used to rate reflexes and responses of newborn babies. Each baby is assigned a score by a medical professional, and the possible values are the integers 0 to 10. A sample of n = 1000 babies born in a certain county is taken and the number with each score is as follows:

Xi = Score 0 1 2 3 4 5 6 7 8 9 10 f i = No. of babies

a. Find the sample mean Apgar score for these 1000 observations (This is grouped data. You will need to use slightly modified formulas for the mean and standard deviation. 11

1

i i i

X f x n (^) =

= ∑ and ( ) ( )

(^11 ) 2 1

i i^1 i

S f X X n

= ∑ − − ).

b. Find the sample standard deviation of these Apgar scores. c. What is the median? d. What is the mode? e. What proportion of the scores is greater than the mean? f. What proportion of scores is more than one standard deviation greater than the mean? g. What proportion of the scores is within one standard deviation of the mean?

2. Using the Apgar data in problem 2, construct a modified box-and-whisker plot, identifying any outliers, if present.
3. In a random experiment, two random variables are generated; namely X 1 and X 2. The following table provides the joint probability density function.

X 2 X 1 0 1 2 0 0.40 0.12 0. 1 0.15 0.08 0. 2 0.10 0.03 0.

a. Find the p.d.f. of the sum of these two variables; i.e. Y = X 1 + X 2. b. How would this p.d.f. change if we divide Y by 2? That is, we find the p.d.f. of the average of X 1 and X 2.

4. Let X and Y be independent random variables with μx = 2, σx = 1, μy = 2, and σy = 3. Find the means and variances of the following quantities.

a. 3X b. X + Y c. X – Y d. 2X + 6Y

5. Bags of concrete mix labeled as containing 100 lb have a population mean weight of 100lb and a population standard deviation of 0.5 lb.

a. What is the probability that the mean weight of a random sample of 50 bags is less than 99.9 lb? b. If the population mean weight is increased to 100.15 lb, what is the probability that the mean weight of a sample of 50 bags will be less than 100 lb?