Growth hormones are often used to increase the weight gain of chickens. In an experiment using chickens, five different doses of growth hormone (, , , , and milligrams) were injected into chickens ( chickens were randomly assigned to each dose), and the subsequent weight gain (in ounces) was recorded. A researcher plots the data and finds that a linear relationship appears to hold. Computer output from a least-squares regression analysis for these data is shown below.
(a) What is the equation of the least-squares regression line for these data? Define any variables you use.
(b) Interpret each of the following in context:
(i) The slope
(ii) The intercept
(iv) The standard error of the slope
(c) Assume that the conditions for performing inference about the slope B of the true regression line are met. Do the data provide convincing evidence of a linear relationship between dose and weight gain? Carry out a significance test at the A level.
(d) Construct and interpret a confidence interval for the slope parameter.
(a) The equation of the least-squares regression line is .
(b) The values are
(v) role="math" localid="1652871593895"
(c) There's sufficient convincing evidence to justify the argument.
(d) There is a chance that the weight gain will be between and . While the does, the ounces climb by milligram.
The general regression line equation
By inserting values the regression line becomes:
With Dose and weight gain.
(i) is the first output. The weight per milligram might increase by ounces.
(ii). In the result , the -intercept is mentioned.
This means that the weight will be ounces if the dosage is milligrams.
(iii) is the output. This means that the average prediction error is ounces.
(iv) This suggests that the population's true slope is on average over all feasible samples.
(v) is written as . This means that the least-square regression line explains of the variance in the variables.
The test statistic is
The P-value is the probability of having the test numbers' value or a more dramatic value. The -value in the row role="math" localid="1652872078633" is represented by a number (or interval) in Table B column title:
The null hypothesis is rejected if the -value is less than or equal to the degree of significance.
Degrees of freedom
Table B, in the row of and the column of , has the important t-value.
The essential t-value may be found in table B in the column and in the row.
The boundaries are
Western lowland gorillas, whose main habitat is the central African continent, have a mean weight of with a standard deviation of . Capuchin monkeys, whose main habitat is Brazil and a few other parts of Latin America, have a mean weight of with a standard deviation of . Both weight distributions are approximately Normally distributed. If a particular western lowland gorilla is known to weigh , approximately how much would a capuchin monkey have to weigh, in pounds, to have the same standardized weight as the lowland gorilla?
(e) There is not enough information to determine the weight of a capuchin monkey.
Park rangers are interested in estimating the weight of the bears that inhabit their state. The rangers have data
on weight (in pounds) and neck girth (distance around the neck in inches) for 10 randomly selected bears. Some
regression output for these data is shown below.
Which of the following represents a 95% confidence interval for the true slope of the least-squares regression line relating the weight of a bear and its neck girth?
Of the 98 teachers who responded, said that they had one or more tattoos.
(a) Construct and interpret a confidence interval for the actual proportion of teachers at the AP institute who would say they had tattoos.
(b) Does the interval in part (a) provide convincing evidence that the proportion of teachers at the institute with tattoos is not (the value cited in the Harris Poll report)? Justify your answer.
(c) Two of the selected teachers refused to respond to the survey. If both of these teachers had responded, could your answer to part (b) have changed? Justify your answer
Which of the following statements about the t distribution with degrees of freedom df is (are) true?
I. It is symmetric.
II. It has more variability than the t distribution with degrees of freedom.
III. As df increases, the t distribution approaches the standard Normal distribution.
(a) I only (c) III only (e) I, II, and III
(b) II only (d) I and II
Some high school physics students dropped a ball and measured the distance fallen (in centimeters) at various times (in seconds) after its release. If you have studied physics, then you probably know that the theoretical relationship between the variables is distance . A scatterplot of the students’ data showed a clear curved pattern. At seconds after release, the ball had fallen centimeters. How much more or less did the ball fall than the theoretical model predicts?
(a) More by centimeters
(b) More by centimeters
(c) No more and no less
(d) Less by centimeters
(e) Less by centimeters
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