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Chapter 12: Q. 3 (page 796)

Inference about the slope 尾of a least-squares regression line is based on which of the following distributions?

(a) The t distribution with n-1degrees of freedom

(b) The standard Normal distribution

(c) The chi-square distribution with n-1degrees of freedom

(d) The t distribution with n-2degrees of freedom

(e) The Normal distribution with mean 渭and standard deviation蟽

Short Answer

Expert verified

Inference about the slope 尾of a least-squares regression line is based on option (d) The tdistribution with n-2degrees of freedom.

Step by step solution

01

Concept introduction 

A regression line is marked in statistics that best govern the relationship of a set of data. In other phrases, it's a line that best describes a data set's trend.

02

Explanation

We must estimate two parameters (the y-intercept 伪and the slope 尾) in order to find the least-squares regression line.

The t-distribution with degrees of freedom equal to the sample size nminus the number of calculated parameters becomes the appropriate distribution.

df=n-2.

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Most popular questions from this chapter

Insurance adjusters are always vigilant about being overcharged for accident repairs. The adjusters suspect that Repair Shop 1quotes higher estimates than Repair Shop 2. To check their suspicion,

the adjusters randomly select 12cars that were recently involved in an accident and then take each of the cars to both repair shops to obtain separate estimates of the cost to 铿亁 the vehicle. The

estimates are given below in hundreds of dollars.


Assuming that the conditions for inference are reasonably met, which of the following signi铿乧ance tests could legitimately be used to determine whether the adjusters鈥� suspicion is correct?

I. A paired ttest

II. A two-sample ttest

III. A t test to see if the slope of the population regression line is 0.

(a) I only

(b) II only

(c) I and III

(d) II and III

(e) I, II, and III

Students in Mr. Handford鈥檚 class dropped a kickball beneath a motion detector. The detector recorded the height of the ball as it bounced up and down several times. Here are the heights of the ball at the highest point on the first five bounces:

Bounce numberHeight (feet)12.24021.62031.23540.95850.756

Here is the scatter plot of the data

(a) The following graphs show the results of two different transformations of the data. Would an exponential model or a power model provide a better description of the relationship between bounce number and height? Justify your answer.

(b) Minitab output from a linear regression analysis on the transformed data of log(height) versus bounce number is shown below. Give the equation of the least-squares regression line. Be sure to define any variables you use.

PredictorCoefSE CoefTPConstant0.453740.0138532.760.000Bounce-0.1171600.004176-28.060.000

S=0.0132043R-Sq=99.6%R-Sq(adj)=99.5%

(c) Use your model from part (b) to predict the highest point the ball reaches on its seventh bounce. Show your work.

(d) A residual plot for the linear regression in part (b) is shown below. Do you expect your prediction part (c) to be too high, too low, or just right? Justify your answer.

Marcella takes a shower every morning when she gets up. Her time in the shower varies according to a Normal distribution with mean 4.5minutes and standard deviation 0.9minutes.

(a) If Marcella took a 7-minute shower, would it be classified as an outlier? Justify your answer.

(b) Suppose we choose 10days at random and record the length of Marcella鈥檚 shower each day. What鈥檚 the probability that her shower time is 7minutes or higher on at least 2of the days? Show your work.

(c) Find the probability that the mean length of her shower times on these 10 days exceeds5 minutes. Show your work

The body鈥檚 natural electrical field helps wounds heal. If diabetes changes this field, it might explain why people with diabetes heal more slowly. A study of this idea compared randomly selected normal mice and randomly selected mice bred to spontaneously develop diabetes. The investigators attached sensors to the right hip and front feet of the mice and measured the difference in electrical potential (in millivolts) between these locations. Graphs of the data for each group reveal no outliers or strong skewness. The computer output below provides numerical summaries of the data26.

The researchers want to know if there is evidence of a significant difference in mean electrical potentials between normal mice and mice with diabetes. Carry out a test using a 5%level of significance and report your conclusion.

A standard deck of playing cards contains 52cards, of which 4are aces and 13are hearts. You are offered a choice of the following two wagers:

I. Draw one card at random from the deck. You win \(10if the card drawn is an ace. Otherwise, you lose \)1.

II. Draw one card at random from the deck. If the card drawn is a heart, you win \(2. Otherwise, you lose \)1.

Which of the two wagers should you prefer?

(a) Wager 1, because it has a higher expected value

(b) Wager 2, because it has a higher expected value

(c) Wager 1, because it has a higher probability of winning

(d) Wager 2, because it has a higher probability of winning

(e) Both wagers are equally favorable.
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