Girls emerge from adolescence feeling quite differently about themselves, acco
several studies.
Suppose a number of randomly selected girls in the Ithaca-Binghamton area
of New York State yielded the following data:
X
y
(Girl's age in years old)
(Self-esteem:
0 to 100 scale)
8
11
14
13
16
72
64
82
53
41
46
1.
2.
3.
4
Construct a scatterplot of the data.
Calculate the values for m and for b (by hand), and then write the least-squares
equation in D=mx+b form.
Use the least-squares regression equation to predict self-esteem for a girl of
age (i) 9 and (ii) 15
Use the two (xv)-noints
calculated in number 3 to lot the least-squares

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!

Girls emerge from adolescence feeling quite differently about themselves, according to several studies. Suppose a number of randomly selected girls in the Ithaca-Binghamton area of New York State yielded the following data

Explore a step-by-step solution to constructing a scatterplot and calculating the least-squares regression line using real data. Learn how to predict self-esteem for different ages based on a linear model. This example is perfect for students in Grades 10-12 studying statistics and linear regression.

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