To find the slope estimate $b_1$ for the linear regression model (predicting Sales from Number of Sales People Working), we use the formula for the slope in simple linear regression:

$b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

Where:

• $n$ is the number of data points (in this case, 10 data points).
• $x$ represents the number of sales people working.
• $y$ represents the sales (in $1000).
• $\sum x$ is the sum of the $x$-values.
• $\sum y$ is the sum of the $y$-values.
• $\sum xy$ is the sum of the product of each pair of $x$ and $y$ values.
• $\sum x^2$ is the sum of the squares of the $x$-values.

Let's first organize the data:

Number of Sales People (x)	Sales (y)
3	11
4	12
6	13
8	14
11	18
11	20
13	20
15	23
16	23
19	26

Now, let's calculate the necessary sums:

1. $\sum x = 3 + 4 + 6 + 8 + 11 + 11 + 13 + 15 + 16 + 19 = 106$
2. $\sum y = 11 + 12 + 13 + 14 + 18 + 20 + 20 + 23 + 23 + 26 = 170$
3. $\sum x^2 = 3^2 + 4^2 + 6^2 + 8^2 + 11^2 + 11^2 + 13^2 + 15^2 + 16^2 + 19^2 = 9 + 16 + 36 + 64 + 121 + 121 + 169 + 225 + 256 + 361 = 1378$
4. $\sum xy = (3 \times 11) + (4 \times 12) + (6 \times 13) + (8 \times 14) + (11 \times 18) + (11 \times 20) + (13 \times 20) + (15 \times 23) + (16 \times 23) + (19 \times 26)$ $= 33 + 48 + 78 + 112 + 198 + 220 + 260 + 345 + 368 + 494 = 2118$

Now, we can substitute these values into the slope formula:

$b_1 = \frac{10(2118) - (106)(170)}{10(1378) - (106)^2}$

Let's compute this:

$b_1 = \frac{21180 - 18020}{13780 - 11236} = \frac{3150}{2554} \approx 1.23$

So, the slope estimate $b_1$ is approximately 1.23.

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Would you like any further details or explanations? Also, here are some related questions you might find useful:

1. What does the slope estimate $b_1$ mean in the context of this problem?
2. How do you calculate the y-intercept ($b_0$) in linear regression?
3. How can you determine the correlation between the number of salespeople and sales using the regression model?
4. What is the importance of calculating the coefficient of determination (R²) in regression analysis?
5. How does increasing the number of salespeople affect sales based on the slope estimate?

Tip: Always double-check your calculations to ensure no errors when performing regression analysis.