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Homework answers / question archive / Every day Cara runs for five miles
Every day Cara runs for five miles. Suppose that the time it takes her to complete the run is a random variable that is normally distributed with a mean of 50 minutes and a standard deviation of 5 minutes. Today Cara ran the five miles in 46 minutes. What is the probability that she will beat her time tomorrow?
A person must score in the upper 2% of the population on an IQ test to qualify for membership in Mensa, the international high-IQ society. If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what score must a person have to qualify for Mensa?
1) Computation of Probability that she will beat her time tomorrow:
Cara beat her time tomorrow means she run faster than today.
That is time taken by her to complete five miles is less than 46 min
Given,
μ = 50 , σ = 5
We convert this to standard normal as
P( X < x) = P( Z < x - μ/σ )
So,
P( X < 46) = P( Z < 46 - 50 / 5)
= P( Z < -0.8)
= 1 - P( Z < 0.8)
= 1 - 0.7881 (Probability calculated from Z table)
= 0.2119
So, Probability that she will beat her time tomorrow is 0.2119
2) Computation of score must a person have to qualify for Mensa:
For top 2% z = 2.05
Mean = 100, sd = 15
x = μ + z*σ
= 100 + 2.05*15
x = 130.75
Hence, a person must score 130.75 or higher to qualify for Mensa.
