Question

The amounts of time per workout an athlete uses a stairclimber are normally​ distributed, with a mean of 24 minutes and a standard deviation of 7 minutes. Find the probability that a randomly selected athlete uses a stairclimber for​
(a) less than 19 ​minutes,
(b) between 24 and 33 ​minutes, and​
(c) more than 40 minutes.

Which event is unusual?

Answer 1

Answer:

(a) The probability that a randomly selected athlete uses a stairclimber for​  less than 19 ​minutes is 0.2388.

(b) The probability that a randomly selected athlete uses a stairclimber for​  between 24 and 33 ​minutes is 0.3997.

(c) The probability that a randomly selected athlete uses a stairclimber for​  more than 40 minutes is 0.0113.

Step-by-step explanation:

We are given that the amounts of time per workout an athlete uses a stairclimber are normally​ distributed, with a mean of 24 minutes and a standard deviation of 7 minutes.

Let X = amounts of time per workout an athlete uses a stairclimber

So, X ~ Normal($\mu=24,\sigma^{2} =7^{2}$)

The z-score probability distribution for the normal distribution is given by;

                               Z  =  $\frac{X-\mu}{\sigma}$  ~ N(0,1)

where, $\mu$ = mean time = 24 minutes

            $\sigma$ = standard deviation = 7 minutes

(a) The probability that a randomly selected athlete uses a stairclimber for​  less than 19 ​minutes is given by  P(X < 19 minutes)

        P(X < 19 min) = P( $\frac{X-\mu}{\sigma}$ < $\frac{19-24}{7}$ ) = P(Z < -0.71) = 1 - P(Z $\leq$ 0.71)

                                                           = 1 - 0.7612 = 0.2388

The above probability is calculated by looking at the value of x = 0.71 in the z table which has an area of 0.7612.

(b) The probability that a randomly selected athlete uses a stairclimber for​  between 24 and 33 ​minutes is given by = P(24 min < X < 33 min)

     P(24 min < X < 33 min) = P(X < 33 min) - P(X $\leq$ 24 min)

     P(X < 33 min) = P( $\frac{X-\mu}{\sigma}$ < $\frac{33-24}{7}$ ) = P(Z < 1.28) = 0.8997

     P(X $\leq$ 24 min) = P( $\frac{X-\mu}{\sigma}$ $\leq$ $\frac{24-24}{7}$ ) = P(Z $\leq$ 0) = 0.50

The above probability is calculated by looking at the value of x = 1.28 and x = 0 in the z table which has an area of 0.8997 and 0.50 respectively.

Therefore, P(24 min < X < 33 min) = 0.8997 - 0.50 = 0.3997

(c) The probability that a randomly selected athlete uses a stairclimber for​  more than 40 minutes is given by  P(X > 40 minutes)

        P(X > 40 min) = P( $\frac{X-\mu}{\sigma}$ > $\frac{40-24}{7}$ ) = P(Z > 2.28) = 1 - P(Z $\leq$ 2.28)

                                                           = 1 - 0.9887 = 0.0113

The above probability is calculated by looking at the value of x = 2.28 in the z table which has an area of 0.9887.

The event of probability that a randomly selected athlete uses a stairclimber for​  more than 40 minutes is unusual because this probability is less than 5% and any even whose probability is less than 5% is said to be unusual.