Leave answers as decimals (not percents). When necessary, round final answers to

Question

Leave answers as decimals (not percents). When necessary, round final answers to 4 decimal places.

4. Bone density can be measured using a DEXA test. Bone density for a 25-year-old woman is normally-distributed with a mean of 956 g/cm2 and a standard deviation of 5.52 g/cm2. Let X be the bone density of a randomly selected 25-year-old woman in g/cm2. a. Choose one: X is a binomial random variable X is a normal random variable b. Find P(x > 965) c Hannah is a 25-year-old woman. The z-score of her bone densit is-0.32 Find Hannah's bone density in g/cm2 If 47 randomly selected 25-year-old women have their bone densities measured, what is the probability that the sample mean bone density for this group will be lower than 955 g/cm2? d.

Explanation / Answer

a) X is a normal random variable.

b) P(X > 965)

= P((X - mean)/sd > (965 - mean)/sd)

= P(Z > (965 - 956)/5.52)

= P(Z > 1.63)

= 1 - P(Z < 1.63)

= 1 - 0.9484

= 0.0516

c) Z = -0.32

or, (X - 956)/5.52 = 0.32

or, X = 0.32 * 5.52 + 956

or, X = 957.77

d) P(X < 955)

= P((X - mean)/(sd/sqrt(n) < (955 - mean)/(sd/sqrt(n))

= P(Z < (955 - 956)/(5.52/sqrt(47)))

= P(Z < -1.24)

= 0.1075

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