# Activity 04/09/21

## Student Name:

## Instructions:
Fill in your name above.  We will work through the following activities as a group in the discussion sections.  You must turn in a completed worksheet with solutions filled in below for credit.  The solutions will be given in the discussion section.

## Activity: Computing z-scores and critical values

### Question 1:

Recall, a general normal confidence interval (with known variance $\sigma^2$) is given of the form:

$$\left(\overline{x} - z_\frac{\alpha}{2}\frac{\sigma}{\sqrt{n}}, \overline{x}+z_\frac{\alpha}{2} \frac{\sigma}{\sqrt{n}}\right)$$
where $z_\frac{\alpha}{2}$ is selected so that 

$$P\left(Z\leq z_\frac{\alpha}{2}\right) = 1 - \frac{\alpha}{2}$$.

The confidence level for the above interval is thus equal to $(1-\alpha)\times 100\%$.

For a normal population with known variance $\sigma^2$, answer the following questions:


What is the confidence level (written as a percentage) for the interval
$$\left(\overline{x} - 2.14\frac{\sigma}{\sqrt{n}}, \overline{x}+2.14 \frac{\sigma}{\sqrt{n}}\right)?$$

Describe how you can compute $\alpha$ using the built-in functions in R.

#### Solution:


## Activity: Computing point estimates and confidence intervals

### Question 2:

The yield of a chemical process is being studied. From previous experience, yield is known to be normally distributed and $\sigma=3$. The past five days of plant operation have resulted in the yields in the following array:
```{r}
my_data <- c(91.6, 88.75, 90.8, 89.95, 91.3)
```

Compute each of the:

1. sample mean
2. sample standard deviation
3. the standard error of the sample mean
4. the sample-estimate for the standard error of the sample mean
5. the 95% two-sided confidence interval for the true mean
6. the 99% two-sided confidence interval for the true mean

#### Solution
