To solve this problem, we will first calculate the mean and standard deviation for both the 100-meter backstroke and the 100-meter butterfly times. Once we have these values, we can choose the correct interpretation.

Step 1: Calculate the Mean for Each Stroke

100-Meter Backstroke Times: 53.7, 54.3, 52.4, 52.4, 51.8

Mean = $\frac{53.7 + 54.3 + 52.4 + 52.4 + 51.8}{5}$

100-Meter Butterfly Times: 52.2, 51.5, 49.9, 51.5, 50.5

Mean = $\frac{52.2 + 51.5 + 49.9 + 51.5 + 50.5}{5}$

Let's compute these means.

Step 2: Calculate the Standard Deviation for Each Stroke

The standard deviation is calculated using the formula:

$\sigma = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n}}$

Where $x_i$ are the individual times, $\bar{x}$ is the mean, and $n$ is the number of data points (in this case, 5). We'll compute this for each stroke as well.

Let me perform the calculations.### Results:

• 100-Meter Backstroke:

  • Mean: $52.92$ seconds
  • Standard Deviation: $0.93$ seconds (rounded to two decimal places)

• 100-Meter Butterfly:

  • Mean: $51.12$ seconds
  • Standard Deviation: $0.82$ seconds (rounded to two decimal places)

Step 3: Choose the Correct Interpretation

The correct interpretation for the mean gold medal time in the context of the problem would be:

Option B: "The typical gold medal time for the 100-meter backstroke in the five recent competitions is 52.92 seconds."

Would you like to explore any more details or have any further questions?

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Relative Questions:

1. How does the standard deviation reflect the consistency of the athletes' performance?
2. How would the mean and standard deviation change if one of the backstroke times was significantly slower or faster?
3. What is the significance of a lower standard deviation in this context?
4. How would adding more data points (e.g., more competition times) affect the calculations of the mean and standard deviation?
5. How can this statistical analysis help in predicting future competition results?

Tip: When interpreting the standard deviation, a smaller value indicates that the times are more consistent across different competitions.