Let's first compute the slope of the least squares line. The slope ($b_1$) of the least squares line can be computed using the following formula:

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

Where:

• $n$ is the number of data points (7 in this case),
• $\sum{xy}$ is the sum of the product of each $x$ and corresponding $y$,
• $\sum{x}$ is the sum of all $x$ values,
• $\sum{y}$ is the sum of all $y$ values,
• $\sum{x^2}$ is the sum of the squares of all $x$ values.

Step 1: Compute the necessary sums

• $x = [178, 176, 174, 166, 172, 172, 174]$
• $y = [58, 55, 54, 47, 52, 53, 55]$

Now let's calculate:

1. $\sum{x} = 178 + 176 + 174 + 166 + 172 + 172 + 174$
2. $\sum{y} = 58 + 55 + 54 + 47 + 52 + 53 + 55$
3. $\sum{xy} = (178 \times 58) + (176 \times 55) + (174 \times 54) + (166 \times 47) + (172 \times 52) + (172 \times 53) + (174 \times 55)$
4. $\sum{x^2} = (178^2) + (176^2) + (174^2) + (166^2) + (172^2) + (172^2) + (174^2)$

I'll calculate these values next.The computed sums are:

• $\sum{x} = 1212$
• $\sum{y} = 374$
• $\sum{xy} = 64832$
• $\sum{x^2} = 209936$

Step 2: Compute the slope ($b_1$) using the formula:

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

Let me calculate this value for you.The slope of the least squares line is $b_1 = 0.882$ (rounded to three decimal places).

Step 3: Find the equation of the least squares line

The equation of the least squares line is given by:

$y = b_0 + b_1x$

Where:

• $b_1$ is the slope we just calculated,
• $b_0$ (the y-intercept) is provided as $-99.2105263157895$.

Thus, the equation of the least squares line is:

$y = -99.211 + 0.882x$

Step 4: Hypothetical regression equation

If the regression equation were $y = -80 + 0.8x$, for $x = 177$, the predicted value of $y$ would be:

$y = -80 + 0.8(177)$

Let me calculate this value for you.For $x = 177$, the hypothetical regression line $y = -80 + 0.8x$ predicts that $y = 61.6$.

Would you like further details on any part of this solution?

Here are some related questions to explore:

1. How do you calculate the y-intercept if it's not given?
2. What are residuals in a regression model?
3. How do you interpret the slope in a real-world context?
4. How do you calculate the coefficient of determination $R^2$?
5. How can you perform a significance test for the slope?

Tip: Always check the assumptions of linear regression before relying on predictions from your model.