Instructor: Colin Grudzien

## Instructions: We will work through the following series of activities as a group and hold small group work and discussions in Zoom Breakout Rooms. Follow the instructions in each sub-section when the instructor assigns you to a breakout room. ## Activities: ### Activity 1: 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 $\sigma^2$, what is the confidence level (written as a percentage) for the interval $$\left(\overline{x} - 2.14\frac{\sigma}{\sqrt{n}}, \overline{x}+2.14 \frac{\sigma}{\sqrt{n}}\right)?$$ ### Activity 2: Computing point estimates and confidence intervals #### Question 1: 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 ### Activity 3: Central limit theorem #### Question 1: Recall, we can generate a random sample of the Poisson distribution with the `rpois` function. Let us set $\lambda=25$ and draw a sample of size `n=20` from this distribution, and compute the sample mean. Let's replicate the process and plot the sample means over $m=50$ replications. Plot the corresponding central limit theorem sampling distribution for the mean along with these values, with a range given over $[20, 30]$.