The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 25 minutes and a standard deviation of 7 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (a) less than 21 minutes, (b) between 25 and 33 minutes, and (c) more than 40 minutes.
(a) The probability that a randomly selected athlete uses a stairclimber for less than 21 minutes is (Round to four decimal places as needed.)
The correct answer and explanation is:
Answers:
(a) The probability that a randomly selected athlete uses a stairclimber for less than 21 minutes is 0.2839 (rounded to four decimal places).
(b) The probability that a randomly selected athlete uses a stairclimber for between 25 and 33 minutes is 0.3735.
(c) The probability that a randomly selected athlete uses a stairclimber for more than 40 minutes is 0.0161.
Explanation:
The problem states that the time an athlete spends on a stairclimber follows a normal distribution with a mean of 25 minutes and a standard deviation of 7 minutes. We use the z-score formula to determine probabilities: Z=X−μσZ = \frac{X – \mu}{\sigma}
where:
- XX is the specific value (e.g., 21, 33, 40 minutes),
- μ\mu (mean) is 25 minutes,
- σ\sigma (standard deviation) is 7 minutes.
(a) Less than 21 minutes:
Z=21−257=−47≈−0.571Z = \frac{21 – 25}{7} = \frac{-4}{7} \approx -0.571
Using the normal distribution table, the probability for Z=−0.571Z = -0.571 is 0.2839.
(b) Between 25 and 33 minutes:
Z1=25−257=0Z_1 = \frac{25 – 25}{7} = 0 Z2=33−257=87≈1.143Z_2 = \frac{33 – 25}{7} = \frac{8}{7} \approx 1.143
Using the normal table,
P(Z1)=0.5000P(Z_1) = 0.5000,
P(Z2)=0.8735P(Z_2) = 0.8735,
So, the probability is 0.8735−0.5000=0.37350.8735 – 0.5000 = 0.3735.
(c) More than 40 minutes:
Z=40−257=157≈2.143Z = \frac{40 – 25}{7} = \frac{15}{7} \approx 2.143
Using the normal table,
P(Z≤2.143)=0.9839P(Z \leq 2.143) = 0.9839,
so P(X>40)=1−0.9839=0.0161P(X > 40) = 1 – 0.9839 = 0.0161.
Thus, the probabilities are:
- 28.39% chance of working out less than 21 minutes.
- 37.35% chance of working out between 25 and 33 minutes.
- 1.61% chance of working out more than 40 minutes.
The graph visually represents these probabilities using shaded areas under the normal curve.