Tom Cruise is reported to be 5'7.8" tall. Male heights are normally distributed with a mean of 70 inches and a standard deviation of 2.8 inches. Find the probability that;
1) A randomly selected male is Tom Cruise's height or smaller.
2) Three randomly selected males are Tom Cruise's height or smaller.
5' 7.8" = 67.8
http://davidmlane.com/normal.html
click on "area from value"
enter 70 into mean
and 2.8 into SD
click on 'below' and enter 67.8
you should get .215999
for the second, do you want the prob that all 3 are smaller?
if so, then prob = (.215999)^3 = appr .01
To find the probability in each scenario, we need to standardize and use the z-score formula. The z-score measures the number of standard deviations an individual value is from the mean.
Let's calculate the z-score for Tom Cruise's height:
1) Probability that a randomly selected male is Tom Cruise's height or smaller:
First, we need to standardize Tom Cruise's height using the z-score formula:
z = (x - μ) / σ
Where:
- x is the value we want to standardize (Tom Cruise's height)
- μ is the mean of the population (70 inches)
- σ is the standard deviation of the population (2.8 inches)
z = (5.78 - 70) / 2.8 ≈ -2.36
Now, we can look up the corresponding area under the standard normal distribution curve for z = -2.36. Using a standard normal distribution table or a calculator, we find that the area to the left of -2.36 is approximately 0.0082.
Therefore, the probability that a randomly selected male is Tom Cruise's height or smaller is approximately 0.0082, or 0.82%.
2) Probability that three randomly selected males are Tom Cruise's height or smaller:
Since we are interested in three males, we need to consider the probability of selecting one male from the population multiple times.
The probability of selecting one male of Tom Cruise's height or smaller is 0.0082 (as calculated in the previous part).
To find the probability of selecting three males of Tom Cruise's height or smaller, we multiply this probability by itself three times:
P(three selected males) = P(one selected male) * P(one selected male) * P(one selected male)
= 0.0082 * 0.0082 * 0.0082
≈ 0.000000531
Therefore, the probability that three randomly selected males are Tom Cruise's height or smaller is approximately 0.000000531, or 0.0000531%.