Terri Vogel, an amateur motorcyde racer, averages 129,71 seconds per 2.5 mile la

Question

Terri Vogel, an amateur motorcyde racer, averages 129,71 seconds per 2.5 mile lap (na seven-lap race) with a standard deviation of 2.25 seconds The distribution of her race times is normally distributed, we are interested in one of her randomly selected laps. In words, define the random variable X the sme (in seconds) per lap the time on seconds) per race o the distance (in miles) each lap the distance miles of each race Give distribution ofX Find percent of her laps that are completed inless than 134 seconds (Round your answer to two decimal places) the

Explanation / Answer

Answer) Mean = 129.71   S.D = 2.28

Part (b) The distribution of X is given as follows

X follows a Normal Distribution with parameters 129.71 and 5.1984

Part (c) We need to find the percentage of lapses that are completed in less than 134 seconds

We need to find P(X < 134)

The Z score formula is Z = (X - Mean) / S.D

P( Z < (134 - 129.71) / 2,28) = P(Z < 1.8816)

Hence, the probability of lapses that are completed within 134 seconds is 0.9699

Hence, the percentage of lapses that are completed within 134 seconds is 96.99%

Part (d) The fastest 5% of her laps are to be determined here

Hence, the probability that the fastest 5% is 1-0.05 = 0.95

The Z score corresponding to this probability of 0.95 is 1.65

Hence, Z = (X - Mean) / S.D

We need to determine X here

      1.65 = (X - 129.71) / 2.28

      X = 1.65*2.28 + 129.71

      X = 133.472

Hence, the fastest 5% of her laps are under 133.472 seconds

Part (e) The middle 75% of her lap times need to be found out

Middle 75% means, on each side of the normal curve there is an equal distribution of 37.5% of the points on each side of the curve

We need to determine two values 1) X1 = Lower 37.5% and 2) X2 = Upper 37.5%

So, X1 is 0.5 - 0.375 = 0.125 and X2 is 0.5+0.375 = 0.875

X1 = 0.125 and X2 = 0.875

We have the formula for Z score as Z = (X - Mean) / S.D

                               Hence, X = Z*S.D + Mean

The Z score for 0.125 is -1.15 and the Z score for 0.875 is 1.15

Hence, X1 = (-1.15*2.28) + 129.71 = 127.088

            X2 = (1.15*2.28) + 129.71 = 132.332

Hence, the middle 75% of lap times of Terri Vogel, are from 127.09 seconds to 132.332 seconds

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