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:
To find the mean and standard deviation of the total time for the relay team, we can use properties of normal distributions and basic statistics.
Given:
| 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 |
Step 1: Total Mean
The mean of the sum of independent variables is simply the sum of their means:
Total Mean
= 50.71 + 54.45 + 48.26 + 44.25
= 197.67 seconds
Step 2: Total Standard Deviation
Since the swimmers’ performances are independent and follow a normal distribution, we can find the total standard deviation by summing the variances (squared standard deviations) and taking the square root.Variancetotal=(0.28)2+(0.22)2+(0.22)2+(0.23)2=0.0784+0.0484+0.0484+0.0529=0.2281\text{Variance}_{\text{total}} = (0.28)^2 + (0.22)^2 + (0.22)^2 + (0.23)^2 = 0.0784 + 0.0484 + 0.0484 + 0.0529 = 0.2281Variancetotal=(0.28)2+(0.22)2+(0.22)2+(0.23)2=0.0784+0.0484+0.0484+0.0529=0.2281
Now take the square root:SDtotal=0.2281≈0.48 seconds\text{SD}_{\text{total}} = \sqrt{0.2281} \approx 0.48 \text{ seconds}SDtotal=0.2281≈0.48 seconds
Final Answer:
- Mean: 197.67 seconds
- Standard Deviation: 0.48 seconds
Explanation:
In a relay race, each swimmer’s time adds to the total. Because their performances are independent and normally distributed, we can treat their times as independent normal random variables. This means that the distribution of the team’s total time is also normal. The mean total time is the sum of individual means and reflects the average performance across all strokes. The standard deviation represents the variability in the total time. Since standard deviations do not simply add, we combine their variances (squares of the standard deviations) to account for the combined uncertainty and then take the square root. This gives us a complete model for the expected performance of the relay team.
