Please see the **detailed homework policy document** for information about homework formatting, submission, and grading.

The above (simulated, and not the same as the previous homework) data shows the relationship between sleep (in hours) and weight (in kilograms) of a random sample of adult males on a particular night. A simple linear regression model was fit to this data. The fitted line is added to the above plot.

```
##
## Call:
## lm(formula = sleep ~ wt, data = sleep_wt_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9405 -0.5596 -0.0905 0.6341 1.5812
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.75851 2.71887 5.06 8.15e-05 ***
## wt -0.06833 0.02969 NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.951 on 18 degrees of freedom
## Multiple R-squared: 0.2274, Adjusted R-squared: 0.1845
## F-statistic: 5.297 on 1 and 18 DF, p-value: 0.03352
```

Some evil professor has hacked `R`

and ruined the output from the `summary()`

function. Use what information is provided to carry out the test

\[ H_0: \beta_1 = 0 \quad \text{vs} \quad H_1: \beta_1 \neq 0 \]

Report:

- The value of the
**test statistic** - The
**p-value**of the test- Provide a single line of
`R`

**code**used to perform this calculation.

- Provide a single line of
- A
**decision**using \(\alpha = 0.05\)

Using only the information provided in Exercise 1, create 95% confidence intervals for \(\beta_0\) and \(\beta_1\).

The above (simulated, and not the same as the previous homework) data shows the relationship between exam scores and sleep (in hours) for a random sample of students in a large statistics course. A simple linear regression model was fit to this data. The fitted line is added to the above plot.

- \((\bar{x}, \bar{y}) = (6.59375, 82.375)\)
- \(n = 16\)
- \(S_{xx} = 17.869375\)
- \(S_{yy} = 1273.75\)
- \(S_{xy} = 81.5375\)
- \(RSS = 901.6964779\)

Use the above statistics to calculate:

- A 99% confidence interval for the mean exam score of students who sleep 7 hours.
- A 99% prediction interval for the exam score of a student who sleeps 7 hours.

The following three plots show:

- The
**data**and**fitted regression**for the`Orange`

data in`R`

. Here we are using the circumference of oranges as the response and the age of tree as the predictor. - A
**fitted versus residuals plot** - A normal
**qq-plot**