Unit 9 test: Inference for Slopes

In regression inference, what does the parameter $\beta$ represent?

A confidence interval for a population slope is based on a random sample of $30$ individuals. How many degrees of freedom should be used to find $t^*$?

The residual plot below comes from a least-squares regression on a random sample of $12$ individuals. Which condition for inference about the slope is most clearly violated?

A random sample of $40$ students is selected from a school of $300$ students to study the relationship between hours of sleep and test scores. Which condition for regression inference is not met?

A random sample of $15$ employees at a large company gives the output below, where $x$ is years of experience and $y$ is hourly wage (dollars).

S = 3.087, R-Sq = 58.7%

Assuming the conditions are met, find the upper endpoint of a $95\%$ confidence interval for the population slope. Round to the nearest thousandth.

Using the output from the previous problem, a $95\%$ confidence interval for the slope is about $(0.158,\ 0.478)$. Which is the best interpretation?

A $90\%$ confidence interval for a population slope, based on a random sample of $12$ individuals, is $(1.05,\ 2.25)$. Find $SE_b$. Round to the nearest thousandth.

All else equal, which change would make a confidence interval for the population slope wider?

A nutritionist wants to know whether there is a negative linear relationship between the sugar content of breakfast cereals ($x$, grams) and their nutrition rating ($y$). Which hypotheses should she test?

Regression output shows a slope of $-1.84$ with a standard error of $0.61$. Find the $t$ statistic for testing $H_0: \beta = 0$. Round to the nearest hundredth.

A rental company takes a random sample of $13$ of its electric scooters and records each scooter's age ($x$, months) and its range on a full charge ($y$, miles). The company suspects that older scooters have shorter ranges.

S = 1.2981, R-Sq = 21.7%

For a test of $H_0: \beta = 0$ versus $H_a: \beta < 0$, what is the P-value?

Assume the conditions are met for the scooter test in the previous problem. At the $\alpha = 0.05$ level, which conclusion is correct?

A $99\%$ confidence interval for the slope relating a city's population density ($x$) to its average commute time ($y$) is $(-0.8,\ 4.2)$. What is the result of a test of $H_0: \beta = 0$ versus $H_a: \beta \ne 0$ at $\alpha = 0.01$?

A new bathroom scale is checked against a certified scale using a random sample of $20$ objects. If the new scale is accurate, the slope relating its reading ($y$) to the certified reading ($x$) should be $1$. The output gives $b = 0.87$ and $SE_b = 0.052$. Find the test statistic for $H_0: \beta = 1$ versus $H_a: \beta \ne 1$.

Regression output for predicting a plant's height (cm) from days since planting shows SE Coef $= 0.042$ in the Days row. Which is the best interpretation of this value?