In a murder trial in Los Angeles, a shoe expert stated that the range of heights of men with a size 12 shoe is 72 inches to 76 inches. Suppose the heights of all men wearing size 12 shoes are normally distributed with a mean of 73.5 inches and a standard deviation of 1 inch. What is the probability that a randomly selected man who wears a size 12 shoe (Round answers to four decimal places.)
(a)Has a height outside the range 72 inches to 76 inches?
(b)Is 74 inches or taller?
(c)Is shorter than 71.5 inches?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
To find the probabilities in this problem, we need to use the concept of the standard normal distribution, also known as the Z-distribution. We will convert the heights into standard deviations (Z-scores) and use a Z-table to find the probabilities.
(a) To find the probability that a randomly selected man has a height outside the range of 72 inches to 76 inches, we need to find the area under the curve beyond these bounds.
First, we convert the values of 72 inches and 76 inches into Z-scores using the formula:
Z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation.
For 72 inches:
Z1 = (72 - 73.5) / 1 = -1.5
For 76 inches:
Z2 = (76 - 73.5) / 1 = 2.5
Now, we need to find the probability of a Z-score being less than -1.5 or greater than 2.5. We consult a Z-table to find these probabilities:
The probability of a Z-score less than -1.5 is approximately 0.0668.
The probability of a Z-score greater than 2.5 is approximately 0.0062.
To find the probability of being outside the range, we add these two probabilities:
P(outside range) = P(Z < -1.5 or Z > 2.5) = P(Z < -1.5) + P(Z > 2.5)
P(outside range) = 0.0668 + 0.0062 = 0.073
Therefore, the probability that a randomly selected man who wears a size 12 shoe has a height outside the range of 72 inches to 76 inches is approximately 0.073.
(b) To find the probability that a randomly selected man who wears a size 12 shoe is 74 inches or taller, we need to find the area under the curve beyond 74 inches.
First, we convert 74 inches to a Z-score:
Z = (74 - 73.5) / 1 = 0.5
Next, we find the probability of a Z-score greater than 0.5 using the Z-table:
P(Z > 0.5) = 1 - P(Z < 0.5)
From the Z-table, we find that P(Z < 0.5) is approximately 0.6915.
Therefore, P(Z > 0.5) = 1 - 0.6915 = 0.3085.
Therefore, the probability that a randomly selected man who wears a size 12 shoe is 74 inches or taller is approximately 0.3085.
(c) To find the probability that a randomly selected man who wears a size 12 shoe is shorter than 71.5 inches, we need to find the area under the curve before 71.5 inches.
First, we convert 71.5 inches to a Z-score:
Z = (71.5 - 73.5) / 1 = -2
Next, we find the probability of a Z-score less than -2 using the Z-table:
P(Z < -2) ≈ 0.0228.
Therefore, the probability that a randomly selected man who wears a size 12 shoe is shorter than 71.5 inches is approximately 0.0228.