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To find the standard deviation of the paired differences, we need to follow these steps:

1. **Calculate the Differences \( d \)**:
   \[
   d = \text{Score (old design)} - \text{Score (new design)}
   \]

2. **Compute the Mean of the Differences \( \bar{d} \)**:
   \[
   \bar{d} = \frac{\sum d_i}{n}
   \]

3. **Find the Sum of Squared Differences from the Mean**:
   \[
   \sum (d_i - \bar{d})^2
   \]

4. **Compute the Standard Deviation**:
   \[
   s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}}
   \]

Now let's apply these steps to the data provided:

### Step 1: Calculate the Differences
| Golfer | Score (old design) | Score (new design) | Difference (\( d \)) |
|--------|--------------------|--------------------|---------------------|
| 1      | 81                 | 78                 | 3                   |
| 2      | 78                 | 82                 | -4                  |
| 3      | 83                 | 78                 | 5                   |
| 4      | 80                 | 76                 | 4                   |
| 5      | 87                 | 92                 | -5                  |
| 6      | 87                 | 83                 | 4                   |
| 7      | 89                 | 88                 | 1                   |
| 8      | 94                 | 89                 | 5                   |

### Step 2: Calculate the Mean of the Differences
\[
\bar{d} = \frac{3 + (-4) + 5 + 4 + (-5) + 4 + 1 + 5}{8} = \frac{13}{8} = 1.625
\]

### Step 3: Calculate the Sum of Squared Differences from the Mean
\[
\sum (d_i - \bar{d})^2 = (3 - 1.625)^2 + (-4 - 1.625)^2 + (5 - 1.625)^2 + (4 - 1.625)^2 + (-5 - 1.625)^2 + (4 - 1.625)^2 + (1 - 1.625)^2 + (5 - 1.625)^2
\]
\[
= 1.8906 + 31.6406 + 11.3906 + 5.6406 + 43.8906 + 5.6406 + 0.3906 + 11.3906 = 111.875
\]

### Step 4: Calculate the Standard Deviation
\[
s_d = \sqrt{\frac{111.875}{7}} = \sqrt{15.9821} \approx 4.0
\]

So, the standard deviation of the paired differences is approximately **4.0**.

Would you like to discuss this further or ask about related topics? Here are some questions to expand on:

1. What is the interpretation of this standard deviation value in context?
2. How would you conduct a hypothesis test with this data?
3. What is the significance of using a paired t-test in this scenario?
4. How do you compute confidence intervals for the mean difference?
5. What assumptions are required for the paired t-test?

**Tip:** Always ensure the data meets the assumptions for statistical tests, such as normality and the paired nature of the data.

A golf club manufacturer claims that golfers can lower their scores by using the manufacturer's newly designed golf clubs. Eight golfers are randomly selected and asked for their most recent scores. After using the new clubs for one month, their new scores are recorded. Is there enough evidence to support the manufacturer's claim? Use a significance level of 0.01 and assume normal distribution.

Evaluate if golfers improved their scores with newly designed golf clubs. This problem involves calculating the standard deviation of paired differences, applying hypothesis testing, and interpreting results using a significance level of 0.01. Key concepts include paired differences and standard deviation.

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