Consider the following data from a sample of n=7.

x 178 176 174 166 172 172 174
y 58 55 54 47 52 53 55

The y-intercept of the least squares line is -99.2105263157895. Compute the slope of the least squares line and enter the equation of the least squares line below. (As always, if you round, make sure you do so correctly, only round your final answer, and keep at least three decimal places.)

The least squares line is y =

Regardless of what answer you actually got, suppose your regression equation were y= -80 + 0.8x.

For x = 177, the hypothetical line predicts the value of y will be

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.

Learn how to calculate the slope and equation of a least squares regression line for a dataset with x-values [178, 176, 174, 166, 172, 172, 174] and y-values [58, 55, 54, 47, 52, 53, 55]. Explore key statistical concepts, including the least squares method and the equation of a straight line. Suitable for high school and undergraduate students.

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