# The amounts of time per workout an athlete uses a stair climber are normally distributed, with a mean of 20 minutes and a standard deviation of 7 minutes. Find the probability that a randomly selected athlete uses a stair climber for (a) less than 16 minutes, (b) between 20 and 28 minutes, and (c) more than 30 minutes.

We can derive the following from the given information:

Time per workout that an athlete uses a stair climber ~ N(mean = 20, sd = 7)

a) We want P(X < 16).

Find the Z score of 16, where Z = (X - mean)/sd = (16 - 20)/7 = -0.57...

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We can derive the following from the given information:

Time per workout that an athlete uses a stair climber ~ N(mean = 20, sd = 7)

a) We want P(X < 16).

Find the Z score of 16, where Z = (X - mean)/sd = (16 - 20)/7 = -0.57 (to 2 decimal places).

Using the standard normal distribution table available here (Table), we find that P(Z < -0.57) = 0.28434

Alternatively, we can use R Software to calculate this probability via the following code:

> pnorm(16,mean=20,sd=7)
[1] 0.2838546

b) We want P(20 < X < 28)

When X1 = 20, Z1 = (X1 - mean)/sd = (20 - 20)/7 = 0

When X2 = 28, Z2 = (X2 - mean)/sd = (28 - 20)/7 = 1.14 (to 2 decimal places).

Thus P(0 < Z < 1.14) = P(Z < 1.14) - P(Z < 0) = 0.87286 - 0.5 = 0.37186 (using tables).

Alternatively, we can use the following code to find this probability via R software:

> pnorm(28,mean=20,sd=7)-pnorm(20,mean=20,sd=7)
[1] 0.373451

c) We want P(X > 30)

When X = 30, Z = (X - mean)/sd = (30 - 20)/7 = 1.43 (to 2 decimal places).

P(Z > 1.43) = 1 - P(Z < 1.43) = 1 - 0.92364 = 0.07636 (from tables).

Alternatively, we can use the following code in R to find this probability:

> pnorm(30,mean=20,sd=7, lower.tail = FALSE)
[1] 0.07656373

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