To find the expected number of defective watches in the total of 10,400 watches based on the sample data, we can use the proportion of defective watches found in the sample.
From the sample of 400 watches, 3 are defective. First, we calculate the proportion of defective watches in the sample:
\[ \text{Proportion of defective watches} = \frac{3}{400} \]
Now, we will use this proportion to estimate the number of defective watches in the total population of 10,400 watches:
\[ \text{Expected number of defective watches} = \text{Total watches} \times \text{Proportion of defective watches} \]
Substituting the values:
\[ \text{Expected number of defective watches} = 10,400 \times \frac{3}{400} \]
Calculating this:
\[ \text{Expected number of defective watches} = 10,400 \times 0.0075 = 78 \]
Thus, the expected number of defective watches in the total of 10,400 watches is approximately 78 after rounding to the nearest whole number.