Performance in the third 100 metres of a 400-metre freestyle swimming race is crucial to the outcome of the race. Based on historical data, the time to complete this third 100 metres among senior swimmers follows an approximate normal distribution with a mean of 110 seconds and a standard deviation of 17 seconds.

(a) Identify the variable of interest and the unit of measurement of the variable in considering the impact of this part of the 400-metre freestyle swimming race.

(b) Based on historical data, what proportion of senior swimmers will take more than 135 seconds to complete the third 100 metres of the 400-metre freestyle event?

B. Based on historical data, approximately 43.06% of senior swimmers will take more than 135 seconds to complete the third 100 meters of the 400-meter freestyle event.

(a) The variable of interest in this scenario is the time taken to complete the third 100 meters of the 400-meter freestyle swimming race. The unit of measurement for this variable is seconds.

(b) To find the proportion of senior swimmers who will take more than 135 seconds to complete the third 100 meters of the race, we need to calculate the area under the normal distribution curve beyond 135 seconds.

Using the given mean (110 seconds) and standard deviation (17 seconds), we can standardize the value of 135 seconds using the z-score formula:

z = (x - μ) / σ

where x is the value (135 seconds), μ is the mean (110 seconds), and σ is the standard deviation (17 seconds).

z = (135 - 110) / 17 = 1.471

We can then look up the proportion associated with this z-score using a standard normal distribution table or a calculator. The proportion represents the area under the curve beyond 135 seconds.

Using a standard normal distribution table, the proportion corresponding to a z-score of 1.471 is approximately 0.4306.

Therefore, based on historical data, approximately 43.06% of senior swimmers will take more than 135 seconds to complete the third 100 meters of the 400-meter freestyle event.

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