A
A researcher collected data on the cholesterol level, C, and the age, A, of 24 people selected at random. Using the data, the researcher calculated the least-squares regression line to be Cˆ=182+2.2A and the standard error of the slope to be 0.38. If the conditions for inference are met, which of the following is closest to the value of the test statistic to test the hypotheses
H_0 : \beta = 0
H_0 : \beta != 0
?
(a) t=0.17
(b) t=0.38
(c) t=0.836
(d) t=2.2
(e) t=5.79
(e) t=5.79
When computing a confidence interval for the slope of a regression line, a plot of the residuals versus the explanatory values can be used to check which of the following conditions?
(A) The variables x and y have a correlation close to 1.
(B) The standard deviation of y does not vary as x varies.
(C) For any fixed value of x, the response y varies according to a Normal distribution.
(D) The observations are independent.
(E) The data come from a well-designed random sample or randomized experiment.
(B) The standard deviation of y does not vary as x varies.
A recent study was done that took a random sample of teenagers and record the number of hours of television they watched (per week) and their GPA (on a 4.0 scale). A 95% confidence interval for the true slope of this regression equation was determined to be (-0.056, -0.012). How can we interpret this confidence interval?
We are 95% confident that each additional hour of watching the television per week is associated with a mean drop in GPA of a value between 0.012 and 0.056.
An agronomist is an expert in soil management and crop production. A certain state hires an agronomist to investigate whether there is a linear relationship between a wheat stalk’s height and the yield of wheat. The agronomist collected data and used the data to test the claim that there is a linear relationship at a significance level of α=0.05�=0.05. The agronomist tested the following hypotheses.
H_0 = \beta_1 = 0
H_a = \beta_1 != 0
The test yielded a p-value of 0.25. What conclusion can we make about the claim?
The null hypothesis is not rejected because 0.25 > 0.05. There is not sufficient evidence to suggest that there is a linear relationship between a wheat stalk’s height and its yield.
Results for a regression analysis comparing arm span to height.
Predictor. Coeff StDev T P
Constant 1.204 0.3981 3.83 0.000
Height 0.115 0.0595 3.553 0.002
s = 0.168 R-Sq = 35.4% R-Sq(adj) = 33.9%
Provided that the conditions for regression inference are satisfied, what is a 95% confidence interval estimate of the slope of the population regression line for predicting handspan from height?
df = 19
invT(.025, 19) = 2.1
0.115±2.1 • 0.0595
To determine the weight of elephants living in the wild, researchers are not able to put them on a scale. Instead, they measure the diameter of the elephant's foot (ft) and apply linear regression to estimate the weight (in pounds). In a sample of 9 elephants, linear regression was performed and the following statistics were obtained:
Predictor Coeff StDev
Constant 154.3. 7.334
Foot Diameter. 275.5. 26.89
s = 23.33
What is the slope of the population regression line with a 95% confidence interval?
b = 275.5
SE_b = 26.89
invT(.025,8) = 2.306
275. +- 2.306 • 26.89
The Wisconsin DNR is in the process of reintroducing elk to the northern Wisconsin area. This process started back in 1995, when 25 elk were first introduced. The population has grown since to, as of 2021, be at about 115. The DNR wants to select a random sample of elk, weigh them and record their age, and then construct a 95% confidence interval to estimate the population slope of the linear regression equation that can predict the weight (in pounds) of the elk based on their age (in years). The result is a confidence interval of 100 ±20.
Assuming that the conditions for inference for the slope of the regression line are met, what is the interpretation for this interval?
We are 95% confident that the mean increase in the weight of an elk for each one-year increase in the age of 'i?eik is between 80 pounds and 120 pounds.
A car retailer wanted to see if there is a linear relationship between overall mileage and the suggested retail price of a car. The retailer collected data on 18 cars of a similar type selected at random and used the data to test the claim that there is a linear relationship. The following hypotheses were used to test the claim.
H_0 = \beta_1 = 0
H_a = \beta_1 != 0
The test yielded a t-value of 2.186 with a corresponding p-value of 0.044. How can we interpret the p-value?
If there is not a linear relationship between overall mileage and the suggested retail price of a car, the probability of observing a test statistic at least as extreme as 2.186 is 0.044.
The critical value for a linear regression t interval with 5 degrees of freedom is calculated. How will the interval size change if the confidence level increases from 90% to 99%, with all other things being equal?
The interval size will be increased by doubling it.
A major credit card company is interested in whether there is a linear relationship between its internal rating of a customer’s credit risk and that of an independent rating agency. The company collected a random sample of 200 customers and used the data to test the claim that there is a linear relationship. The following hypotheses were used to test the claim.
H_0 = \beta_1 = 0
H_a = \beta_1 != 0
The test yielded a t-value of 3.34 with a corresponding p-value of 0.001. How can we interpret of the p-value?
If the null hypothesis is true, the probability of observing a test statistic at least as extreme as 3.34 is 0.001.
A 95 percent confidence interval for the slope of the regression line relating the number of grams of sugars and the number of calories per 100-gram sample of various breads is given by (1.334, 4.565). The confidence interval is based on a random sample of n types of bread. A check of the conditions for inference on the slope shows they are reasonably met. What is an interpretation of the interval?
We are 95% confident that the true slope of the regression line relating grams of sugars and calories
per 100-gram sample of various breads is between 1.334 and 4.565.
A popular musician believes an increase in the number of times songs are listened to via a streaming service leads to an increase in recording sales. The musician’s recording company selected 50 songs at random and used the data to test the claim that there is a positive linear relationship between the number of times a song is listened to and recording sales. The following hypotheses were used to test the claim.
H_0 = \beta_1 = 0
H_a = \beta_1 != 0
The test yielded a t-value of 1.592 with a corresponding p-value of 0.059. How can we interpret the p-value?
If the null hypothesis is true, the probability of observing a test statistic of 1.592 or greater is 0.059.