In a relay event; four swimmers swim 100 yards each Swimmer Mean SD using a different stroke A college team looks at the times for the swimmers and creates a model based on the (backstroke) 50.71 0.28 assumptions that the swimmers’ performances are 2 (breaststroke) 54.45 0.22 independent; each swimmer’s times follow a normal model and the means and standard deviations of the 3 (butterfly) 48.26 0.22 times in seconds are as shown in the table (freestyle) 44.25 0 23 What are the mean and the standard deviation for the relay teams total time in this event? (Round to two decimal places as needed
The Correct Answer and Explanation is:
Mean and Standard Deviation of the Total Relay Time
We are given that each swimmer’s time is normally distributed, and their performances are independent. Here are the individual times:
| Stroke | Mean (seconds) | SD (seconds) |
|---|---|---|
| Backstroke | 50.71 | 0.28 |
| Breaststroke | 54.45 | 0.22 |
| Butterfly | 48.26 | 0.22 |
| Freestyle | 44.25 | 0.23 |
Total Mean Time:
The mean of the total time is the sum of the means of the individual times:Total Mean=50.71+54.45+48.26+44.25=197.67 seconds\text{Total Mean} = 50.71 + 54.45 + 48.26 + 44.25 = 197.67 \text{ seconds}Total Mean=50.71+54.45+48.26+44.25=197.67 seconds
Total Standard Deviation:
Since the swimmers’ performances are independent, the variances add. So we compute the variance of each swimmer’s time, sum them, then take the square root:Variance=0.282+0.222+0.222+0.232=0.0784+0.0484+0.0484+0.0529=0.2281\text{Variance} = 0.28^2 + 0.22^2 + 0.22^2 + 0.23^2 = 0.0784 + 0.0484 + 0.0484 + 0.0529 = 0.2281Variance=0.282+0.222+0.222+0.232=0.0784+0.0484+0.0484+0.0529=0.2281Standard Deviation=0.2281≈0.48 seconds\text{Standard Deviation} = \sqrt{0.2281} \approx 0.48 \text{ seconds}Standard Deviation=0.2281≈0.48 seconds
Final Answer:
- Mean Total Time: 197.67 seconds
- Standard Deviation: 0.48 seconds
Explanation:
In relay events such as swimming, each team member contributes to the final result by completing a leg of the race. In this case, four swimmers each swim 100 yards using different strokes. The total relay time is the sum of all four individual times. To analyze and model the total time statistically, we assume that each swimmer’s time follows a normal distribution with a known mean and standard deviation. Moreover, since each swimmer’s performance is considered independent of the others, the rules of probability allow us to add their means and variances separately.
The expected or average total time is calculated by simply summing the mean times of all four strokes. This provides the most likely average finish time for the team across many races. Variance, a measure of spread or variability, is treated differently. Because the performances are independent, we add the variances (not the standard deviations) of each swimmer’s time to get the total variance. Taking the square root of this total variance yields the standard deviation of the relay team’s total time.
This method reflects both the central tendency and the variability of the team’s performance. A smaller standard deviation suggests consistent performances among the swimmers, while a larger one would indicate more variability. Understanding these figures helps coaches, analysts, and statisticians make informed predictions and decisions in competitive settings.
