IMSE 440 - ci & pi

class: center, middle

## IMSE 440 
## Applied Statistical Models in Engineering

<br/>

## Simple linear regression

## Confidence & prediction intervals

[ISLR book](https://www.statlearning.com): Chapter 3.2.2

---
# Why do we develop statistical models?

- Make inferences

- Is there a relationship? 
  - How strong is the relationship?
  - Which variable(s) are of most importance?
  - Is there synergy among variables?

- Make predictions

- What is the predicted sales in a new market if we spend 150 in TV ads?
  - How confident are we with the predictions?

---

# Making predictions

Once we developed a model (i.e., estimated the model parameter from data), we can make predictions with

$$\hat{y}=\hat{\beta}_0+\hat{\beta}_1x^*$$

$$\text{where $x^*$ is any given value.}$$

.center[![:scale 40%](./images/xyplane_1.png)]

---

.center[![:scale 80%](images/tv_sales.png)]

What is the 95% confidence interval of the .red[average] sales for .red[all] markets that have a TV budget of 150?

---

# How confident are we w/ the predictions?

$$\small{
  \begin{aligned}
    \hat{y}&=\hat{\beta}\_0+\hat{\beta}\_1 x^\* \\\
    \\\
    \hat{Y}_{mean}&=\hat{B}\_0+\hat{B}_1 x^\* \\\
  \end{aligned}
  }
  $$

Previously, we showed the sample .red[slope] estimator

$$\small{
\begin{aligned}
  \hat{B}\_1&=\frac{\sum (x\_i-\bar{x})Y\_i}{S\_{xx}} \\\
  \hat{B}\_1&\sim \text{N}(\beta\_1, \frac{\sigma^2}{S\_{xx}})
\end{aligned}
}
$$

The sample .green[intercept] estimator

$$\small{\hat{B}_0}=\bar{Y}-\bar{x}\hat{B}_1$$

---

$$\small{\hat{B}_0}=\bar{Y}-\bar{x}\hat{B}_1$$

$$\small{\text{E}[\hat{B}_0]=?\;\;\;\;\text{var}(\hat{B}_0)=?}$$

---
$$\small{\hat{Y}_{mean}=\hat{B}\_0+\hat{B}_1 x^\*}$$

---
# CI for the mean response at $x=x^*$

$$\small{
  \begin{aligned}
  \\\
  \hat{Y}\_{mean}&=\hat{B}\_0+\hat{B}\_1 x^\* \\\
  \\\
  \text{E}\big[\hat{Y}\_{mean}\big]&=\beta\_0+\beta\_1 x^* \\\
  \\\
  \text{SE}\big(\hat{Y}\_{mean}\big)&\approx\text{RSE}\sqrt{\frac{1}{n}+\frac{(x-\bar{x})^2}{S\_{xx}}} \\\
  \end{aligned}
}$$

The CI for the mean response at $x=x^*$ is

$$\small{\hat{\beta}\_0+\hat{\beta}\_1 x^* \pm t\_{\frac{\alpha}{2}, n-2}\cdot}\text{SE}\big(\hat{Y}\_{mean}\big)$$

---

.center[![:scale 80%](images/tv_sales.png)]

A new market is opening next month. 
It has a TV budget of 150.

What is the 95% confidence interval of the predicted sales of this particular new market?

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# Prediction interval (PI)

A confidence interval for a .red[single, particular] response at $x=x^*$

<br>

$$
\begin{aligned}
\hat{Y}&=\hat{B}\_0+\hat{B}\_1x^\*+\epsilon \\\
\\\
&=\hat{Y}\_{\text{mean}}+\epsilon
\end{aligned}
$$