The table shows the​ 100-meter backstroke and the​ 100-meter butterfly gold medal times​ (in seconds) for five recent competitions. Complete parts​ (a) and​ (b) below. 
​100-Meter Backstroke
​100-Meter Butterfly
 
53.7
52.2
54.3
51.5
52.4
49.9
52.4
51.5
51.8
50.5
Question content area bottom
Part 1
a. Find and interpret​ (report in​ context) the mean and standard deviation of the gold medal times for each stroke. Select the correct interpretation of the mean for the​ 100-meter backstroke below and fill in the answer box within your choice. 
​(Type an integer or decimal rounded to the nearest hundredth as​ needed.)
A.
The typical gold medal time for the​ 100-meter backstroke is 
  
enter your response here seconds.
B.
The typical gold medal time for the​ 100-meter backstroke in the five recent competitions is 
  
enter your response here seconds.
C.
The typical time for the​ 100-meter backstroke in the five recent competitions is 
  
enter your response here seconds.
D.
The typical gold medal time for the backstroke in the five recent competitions is 
  
enter your response here seconds.

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.

The table shows the​ 100-meter backstroke and the​ 100-meter butterfly gold medal times​ (in seconds) for five recent competitions. Complete parts​ (a) and​ (b) below. 
​100-Meter Backstroke 
​100-Meter Butterfly 
53.7, 52.2, 54.3, 51.5, 52.4, 49.9, 52.4, 51.5, 51.8, 50.5
Question content area bottom 
Part 1
a. Find and interpret​ (report in​ context) the mean and standard deviation of the gold medal times for each stroke. Select the correct interpretation of the mean for the​ 100-meter backstroke below and fill in the answer box within your choice. 
​(Type an integer or decimal rounded to the nearest hundredth as​ needed.)
A. The typical gold medal time for the​ 100-meter backstroke is enter your response here seconds.
B. The typical gold medal time for the​ 100-meter backstroke in the five recent competitions is enter your response here seconds.
C. The typical time for the​ 100-meter backstroke in the five recent competitions is enter your response here seconds.
D. The typical gold medal time for the backstroke in the five recent competitions is enter your response here seconds.

Learn how to find and interpret the mean and standard deviation of 100-meter backstroke and butterfly gold medal times from recent competitions. This problem covers key statistical concepts suitable for students in Grades 9-12.

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