A research team wants to estimate the standard deviation of time spent on social media by teenagers. Find the minimum sample size needed to be $99\%$ confident that the sample standard deviation $s$ is within $3\%$ of the population standard deviation $\sigma$.

Listed below is a simple random sample of the number of times caffeine (e.g., coffee, energy drinks) was mentioned during episodes of a teen drama series.

$4,5,6,3,7,5,4,6,8,5$

Are the conditions met for constructing a $95\%$ confidence interval estimate for the population mean number of caffeine mentions per episode? If yes, compute the $95\%$ confidence interval for the mean.

A quality control engineer collected a simple random sample of 10 bolts produced in a batch to assess their lengths (in millimeters). The sample has the following statistics:

Sample mean $\left(\bar{x}\right)=24.07\text{ mm}$

Margin of error (ME) for a $90\%$ confidence interval $=0.11\text{ mm}$

Construct the $90\%$ confidence interval for the true mean length of bolts produced in the first batch using the given statistics. Another confidence interval for a sample of bolts from a second batch is reported as $(23.95\text{ mm},24.25\text{ mm})$. How does the result compare to the confidence interval of the second batch? Explain possible reasons for differences in their widths.

A nutritionist uses regression to predict weight loss (in pounds) after six weeks based on daily calorie intake. For a daily intake of $1800$ calories, the $95\%$ prediction interval for weight loss is $(3.5,8.9)$ pounds. What does this interval mean?

Suppose a population has a mean of $3200$ and a standard deviation of $8$. If random samples of size $1600$ are selected, what are the mean and standard deviation of the sampling distribution of the sample means?

A population has a mean of $50$ and a standard deviation of $4$. If a random sample of size $100$ is taken, what is the probability that the sample mean is greater than $50.7$? Would this sample mean be considered unusual?