Let's break down the problem into several steps to answer each question clearly.

1. Scatterplot of the data

We are given the data for the girl's age (X) and self-esteem scores (Y):

Age (X)	Self-Esteem (Y)
8	72
11	64
14	82
13	53
16	41

I will generate the scatterplot for this data.

2. Calculate the values for the slope (m) and intercept (b) by hand

The formula for the least-squares regression line is:

$y = mx + b$

Where:

• $m$ is the slope of the line, and
• $b$ is the y-intercept.

The formulas to calculate $m$ and $b$ are:

$m = \frac{n\sum{XY} - \sum{X}\sum{Y}}{n\sum{X^2} - (\sum{X})^2}$

$b = \frac{\sum{Y} - m\sum{X}}{n}$

Let’s compute these values step-by-step.

• $n = 5$ (since there are 5 pairs of data points).

• First, calculate the necessary sums:

  $\sum X = 8 + 11 + 14 + 13 + 16 = 62$ $\sum Y = 72 + 64 + 82 + 53 + 41 = 312$ $\sum XY = (8 \times 72) + (11 \times 64) + (14 \times 82) + (13 \times 53) + (16 \times 41) = 576 + 704 + 1148 + 689 + 656 = 3773$ $\sum X^2 = 8^2 + 11^2 + 14^2 + 13^2 + 16^2 = 64 + 121 + 196 + 169 + 256 = 806$

Now, substitute these values into the formula for $m$:

$m = \frac{5(3773) - (62)(312)}{5(806) - (62)^2} = \frac{18865 - 19344}{4030 - 3844} = \frac{521}{186} \approx 2.80$

Now for $b$:

$b = \frac{312 - 2.80(62)}{5} = \frac{312 - 173.6}{5} = \frac{138.4}{5} = 27.68$

So the equation of the least-squares regression line is:

$y = 2.80x + 27.68$

3. Predict self-esteem for ages 9 and 15

Now we can use the regression equation $y = 2.80x + 27.68$ to predict the self-esteem for a girl aged 9 and 15.

(i) Age 9:

$y = 2.80(9) + 27.68 = 25.2 + 27.68 = 52.88$

(ii) Age 15:

$y = 2.80(15) + 27.68 = 42 + 27.68 = 69.68$

So, the predicted self-esteem is approximately:

• For a 9-year-old girl: 52.88
• For a 15-year-old girl: 69.68

4. Plot the least-squares regression line

Using the two points calculated in step 3, the line passes through:

• (9, 52.88)
• (15, 69.68)

I will now generate the scatterplot with this regression line.

Let me compute and display the scatterplot and lineIt seems like I can’t do more advanced data analysis right now. Please try again later.

You can use the equation $y = 2.80x + 27.68$ to manually plot the regression line on a scatterplot. If you have further questions, feel free to ask!

Would you like details on any of the steps or have more questions?

Here are five questions to consider:

1. How is the slope $m$ related to the trend in the data?
2. Why is the least-squares regression method preferred for such data?
3. How do changes in age (X) affect the prediction of self-esteem (Y)?
4. Can this model predict self-esteem accurately for ages outside the given range?
5. What assumptions do we make when using least-squares regression?

Tip: Always check the residuals to see how well your regression model fits the data!