In an athletics competition, there were a number of heats of the 1500 m race. In the heats, the times that it took the runners to complete the 1500 m were approximately normally distributed, with a mean time of 225 seconds and a standard deviation of 12 seconds. (i) Find the percentage of runners in these heats who took more than 240 seconds to run the 1500 m. (ii) The 20% of runners with the fastest times qualified for the final. Assuming the race times were normally distributed as described above, work out the time needed to qualify for the final, correct to the nearest second. (b) Sally takes part in a number of different races in the competition. The probability that she makes a false start in any given race is 5%. Find the probability that she makes her first false start in her fourth race. Give your answer correct to 4 decimal places.

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In an athletics competition, there were a number of heats of the 1500 m race - Leaving Cert Mathematics - Question 10 - 2022

Question 10

In an athletics competition, there were a number of heats of the 1500 m race. In the heats, the times that it took the runners to complete the 1500 m were approximat... show full transcript

Worked Solution & Example Answer:In an athletics competition, there were a number of heats of the 1500 m race - Leaving Cert Mathematics - Question 10 - 2022

Step 1

Find the percentage of runners in these heats who took more than 240 seconds to run the 1500 m.

96%

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Answer

To find the percentage of runners who took more than 240 seconds, we first need to calculate the z-score for 240 seconds using the formula:

$z = \frac{x - \mu}{\sigma}$

where:

$x = 240$ seconds
$\mu = 225$ seconds (mean)
$\sigma = 12$ seconds (standard deviation)

Substituting the values gives:

$z = \frac{240 - 225}{12} = \frac{15}{12} = 1.25$

Next, we look up the z-score of 1.25 in the standard normal distribution table, which gives us approximately 0.8944. This value represents the percentage of runners that took less time than 240 seconds:

$P(X < 240) = 0.8944$

To find the percentage of runners who took more than 240 seconds:

$P(X > 240) = 1 - P(X < 240) = 1 - 0.8944 = 0.1056$

Thus, the percentage is:

$0.1056 \times 100 \approx 10.56\%$

Step 2

The 20% of runners with the fastest times qualified for the final.

99%

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Answer

To determine the time needed to qualify for the final for the fastest 20% of runners, we need to find the z-score corresponding to the 80th percentile (as the top 20% corresponds to the bottom 80% of times). From the z-table, we find:

$z \approx 0.84$

Using the z-score formula:

$x = \mu + z \cdot \sigma$

we can substitute the values:

$x = 225 + 0.84 \cdot 12 = 225 + 10.08 = 235.08$

Rounding this to the nearest second gives:

$235 \text{ seconds}$

Step 3

Find the probability that she makes her first false start in her fourth race.

96%

101 rated

Answer

The probability of making a false start in any given race is 5%, or 0.05. Therefore, the probability of not making a false start is:

$P = 1 - 0.05 = 0.95$

To find the probability that Sally makes her first false start in her fourth race, we need to consider that she must not make a false start in the first three races and then make her first false start in the fourth race:

$P = (0.95)^3 \times (0.05) \approx 0.4286$

So the answer, correct to 4 decimal places, is:

$0.0429$

In an athletics competition, there were a number of heats of the 1500 m race - Leaving Cert Mathematics - Question 10 - 2022

Worked Solution & Example Answer:In an athletics competition, there were a number of heats of the 1500 m race - Leaving Cert Mathematics - Question 10 - 2022

Find the percentage of runners in these heats who took more than 240 seconds to run the 1500 m.

The 20% of runners with the fastest times qualified for the final.

Find the probability that she makes her first false start in her fourth race.

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