Why is a prediction interval generally wider than a confidence interval for the mean response?

A mechanic is studying the relationship between the weight of a car ($x$, in hundreds of pounds) and its fuel efficiency ($y$, in $\text{mpg}$). The data collected is shown below:

Predict the fuel efficiency ($\text{mpg}$) for a car that weighs $32.5$ hundred pounds. Consider $R^2=1$.

Given x values: 2, 4, 6, 8, what is x̄ and Σx²?

Consider the relationship between $x$ and $y$ provided in the table below:

Predict the value of $y$ when $x=0.5$.

Given t = 2.228, s = 40, n = 8, x₀ = 10, x̄ = 8, Σx = 64, Σx² = 704, calculate the margin of error for the prediction interval at x₀ = 10. Use the formula:E = t * s * sqrt[1 + 1/n + (x₀ - x̄)² / (n*Σx² - (Σx)²)]

If a prediction interval is very wide, what does this suggest about the regression model or data?

Researchers analyzed the relationship between the average price of a used car (in units of thousands of dollars) and the average price of a new car (in units of thousands of dollars) over $12$ years. The regression equation is $\hat{y}=5.2+0.6x$, with $\bar{x}=28$, $\bar{y}=21$, $\sum (x_i-\bar{x}{)}^2=144$, and $s_e=1.8$. Find the $95\%$ prediction interval for the average used car price when the new car price is $\$32,000$. (Use $t_{10,0.025}=2.228$).

How does a prediction interval differ from a confidence interval in regression analysis?

A study examined the relationship between tire width (in units of millimeters) and braking distance (in units of meters) for a set of sports cars. For a tire width of $245$ millimeters, the statistical software output produces a $95$ percent prediction interval for braking distance as $39.7$ to $49.3$ meters. What is the $95\%$ prediction interval estimate for braking distance for a car with $245$-millimeter-wide tires, and what does this interval mean?

A small business owner suspects a link between employees' daily coffee consumption and their daily productivity rating.

Predict the mean productivity rating for all employees who drink $3.5$ cups of coffee per day.