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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 12: Q. 3 (page 787)

Oil and residuals Researchers examined data on the depth of small defects in the Trans-Alaska Oil Pipeline. The researchers compared the results of measurements on $100$ defects made in the field with measurements of the same defects made in the laboratory. The figure shows a residual plot for the least-squares regression line based on these data. Explain why the conditions for performing inference about the slope $β 1$ of the population regression line are not met.

Short Answer

Expert verified

Equal variance requirement has not been met.

Step by step solution

01

Given information:

We have been given that:

A residual plot for the least-squares regression line based on the data.

02

Explanation:

Conditions for performing inference:

$1 .$ Linearity

$2 .$ Independence

$3 .$ Equal Variance

$4 .$ Normality

$5 .$ Random

According to the residual plot for the least-squares regression line equal variance condition has not been met.

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Most popular questions from this chapter

Lamb’s quarters is a common weed that interferes with the growth of corn. An agriculture researcher planted corn at the same rate in $16$ small plots of ground and then weeded the plots by hand to allow a fixed number of lamb’s quarters plants to grow in each meter of cornrow. The decision on how many of these plants to leave in each plot was made at random. No other weeds were allowed to grow. Here are the yields of corn (bushels per acre) in each of the plots:

Here is some computer output from a least-squares regression analysis of these data. Do these data provide convincing evidence at the $α = 0.05$ level that more lamb’s quarters reduce corn yield?

$Predictor Coef SE Coef T P Constant 166.483 2.725 61.11 0.000 Weeds per - 1.0987 0.5712 - 1.92 0.075 meter S = 7.97665 R - Sq = 20.9 % R - Sq (adj) = 15.3 %$

Killing bacteria Expose marine bacteria to X-rays for time periods from $1$ to $15$ minutes. Here is a scatterplot showing the number of surviving bacteria (in hundreds) on a culture plate after each exposure time:

a. Below is a scatterplot of the natural logarithm of the number of surviving bacteria versus time. Based on this graph, explain why it would be reasonable to use an exponential model to describe the relationship between the count of bacteria and the time.

b). Here is the output from a linear regression analysis of the transformed data. Give the equation of the least-squares regression line. Be sure to defne any variables you use.

c. Use your model to predict the number of surviving bacteria after $17$ minutes.

Yahtzee $(5.3, 6.3)$ In the game of Yahtzee, $5$ six-sided dice are rolled simultaneously. To get a Yahtzee, the player must get the same number on all $5$ dice.

a. Luis says that the probability of getting a Yahtzee in one roll of the dice is ${(\frac{1}{6})}^{5}$ . Explain why Luis is wrong.

b. Nassir decides to keep rolling all $5$ dice until he gets a Yahtzee. He is surprised when he still hasn’t gotten a Yahtzee after $25$ rolls. Should he be? Calculate an appropriate probability to support your answer

Do taller students require fewer steps to walk a fixed distance? The scatterplot shows the relationship between x=height (in inches) and y=number of steps required to walk the length of a school hallway for a random sample of 36 students at a high school.

A least-squares regression analysis was performed on the data. Here is some computer output from the analysis

a. Describe what the scatterplot tells you about the relationship between height and the number of steps.

b. What is the equation of the least-squares regression line? Define any variables you use.

c. Identify the value of each of the following from the computer output. Then provide an interpretation of each value.

i. $b_{0}$

ii. $b_{1}$

iii. s

iv. $S E b_{1}$

Sam has determined that the weights of unpeeled bananas from his local store have a mean of $116 g r a m s$ with a standard deviation of $9 g r a m s$ . Assuming that the distribution of weight is approximately Normal, to the nearest gram, the heaviest $30 %$ of these bananas weigh at least how much?

a. $107 g$

b. $121 g$

C. $111 g$

d. $125 g$

e. $116 g$

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