# Worksheet 23: Estimation for quantitative variables

Your name:

Your student ID number:

## Recap

1. Suppose you surveyed $n=100$ Stanford students and $55$ of them say they support the proctoring pilot.
  
    a. What is the estimate $\hat{\pi}_n$?

    b. What is the standard deviation of $\hat{\pi}_n$?
    
    c. What is a 68% confidence interval for $\pi$?




## Quantitative variables

2. How does the sample size $n$ effect the distribution of the sample mean $\hat{\mu}_n$?



3. What do you notice about the distribution of $\hat{\mu}_n$?



4. Suppose that $n=100$ and $\hat{\sigma}_x=200$. What is the standard deviation of $\hat{\mu}_n$?



5. Suppose that $\hat{\mu}_n=310$ and the standard deviation of $\hat{\mu}_n$ is $20$. What is a 95% confidence interval for $\mu$?

 



## Mini crosswords

6. The sample mean of the difference in crossword times is $-7.3$ seconds. Is this evidence that Clikey are better than Andel?



7. The sample size is $n=33$ and the standard deviation of $x_1,\ldots,x_n$ is $\hat{\sigma}_x= 75$ seconds. What is a confidence interval for $\mu$?



8. Suppose that $\mu = -5$. How large does $n$ need to be so that a 95% confidence interval centered at $-5$ will only include negative numbers? (Assume that $\sigma_x = 75$)

 

## Error bars

9. Confidence intervals are often represented visually as error bars to represent uncertainty. How would design a visualization that emphasizes uncertainty?



## Confidence intervals theory

10. Write down the formula for a $1-\alpha$ confidence interval for $\mu$: