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The times taken by a group of runners to finish a marathon are normally distributed with a mean of 138 minutes and a standard deviation of 24 minutes.

(a) Find the total number of runners in the marathon, given that 6,520 runners finished with a time between 114 minutes and 186 minutes.

(b) Find the number of runners who finished the marathon with a time less than 66 minutes.

Answer :

Let's tackle this problem using the properties of the normal distribution.

Given:


  • Mean ([tex]\mu[/tex]) = 138 minutes

  • Standard Deviation ([tex]\sigma[/tex]) = 24 minutes

  • Number of runners between 114 min and 186 min = 6520


(a) Find the total number of runners in the marathon.

The goal here is to find how many runners completed the marathon. The time range of 114 to 186 minutes corresponds to a segment of the normal distribution.


  1. Calculate the Z-scores for 114 minutes and 186 minutes.

    The Z-score formula is:
    [tex]Z = \frac{X - \mu}{\sigma}[/tex]


    • For 114 minutes:
      [tex]Z_{114} = \frac{114 - 138}{24} = -1.00[/tex]


    • For 186 minutes:
      [tex]Z_{186} = \frac{186 - 138}{24} = 2.00[/tex]




  2. Find the probability for these Z-scores using the standard normal distribution table (or a calculator).


    • Probability for [tex]Z < -1.00[/tex] is approximately 0.1587.

    • Probability for [tex]Z < 2.00[/tex] is approximately 0.9772.



  3. Calculate the proportion of runners who finished between 114 and 186 minutes.

    The proportion is given by:
    [tex]P(114 < X < 186) = P(Z < 2.00) - P(Z < -1.00) = 0.9772 - 0.1587 = 0.8185[/tex]


  4. Determine total number of runners ([tex]N[/tex]).

    Given that 8,185 (0.8185 of total runners) finished within the range:
    [tex]0.8185 \times N = 6520[/tex]

    Solving for [tex]N[/tex]:
    [tex]N = \frac{6520}{0.8185} \approx 7967[/tex]



So, approximately 7967 runners participated in the marathon.

(b) Find the number of runners who finished the marathon with time less than 66 minutes.

  1. Calculate the Z-score for 66 minutes.
    [tex]Z_{66} = \frac{66 - 138}{24} = -3.00[/tex]


  2. Find the probability of [tex]Z < -3.00[/tex].

    From the standard normal distribution table, the probability [tex]P(Z < -3.00)[/tex] is approximately 0.0013.


  3. Calculate the number of runners with times less than 66 minutes.

    [tex]\text{Number of runners} = 0.0013 \times 7967 \approx 10[/tex]



Therefore, approximately 10 runners finished the marathon in less than 66 minutes.

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