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A well-known psychologist has established what she calls her Generalized Anxiety (GA) scale. The GA scale, which is a scale from 0 to 10, measures the "gene anxiety" of an individual, with higher GA scores corresponding to more anxiety. We'd like to make predictions about individuals' sleep behavior based on their GA scores. We've collected bivariate data that give the GA score (denoted by x) and the number of hours of sleep last night (denoted by y) for each of the 11 adults participating in a study. The least-squares regression equation for our dat is y=8.92-0.25x. We have used this equation to predict tonight's sleep time for a woman whose GA score is 6.2. We're now interested in both a prediction interval for her sleep time and a confidence interval for the mean sleep time of individuals with her GA score. We have computed the following for our data. • mean square error (MSE)≈ 0.601 (6.2-x)² 11 + 11 Σ(x₂-x)² i=1 Based on this information, and assuming that the regression assumptions hold, answer the questions below. (If necessary, consult a list of formulas.) ≈ 0.1110, where x₁, x₂, ..., X₁1 denote the GA scores in the sample, and x denotes their mean (a) What is the 99% confidence interval for the mean sleep time (in hours) when the GA score is 6.2? (Carry your intermediate computations to at least four decimal places, and round your answer to at least two decimal places.) Lower limit: Upper limit: X

A well-known psychologist has established what she calls her Generalized Anxiety (GA) scale. The GA scale, which is a scale from 0 to 10, measures the "gene anxiety" of an individual, with higher GA scores corresponding to more anxiety. We'd like to make predictions about individuals' sleep behavior based on their GA scores. We've collected bivariate data that give the GA score (denoted by x) and the number of hours of sleep last night (denoted by y) for each of the 11 adults participating in a study. The least-squares regression equation for our dat is y=8.92-0.25x. We have used this equation to predict tonight's sleep time for a woman whose GA score is 6.2. We're now interested in both a prediction interval for her sleep time and a confidence interval for the mean sleep time of individuals with her GA score. We have computed the following for our data. • mean square error (MSE)≈ 0.601 (6.2-x)² 11 + 11 Σ(x₂-x)² i=1 Based on this information, and assuming that the regression assumptions hold, answer the questions below. (If necessary, consult a list of formulas.) ≈ 0.1110, where x₁, x₂, ..., X₁1 denote the GA scores in the sample, and x denotes their mean (a) What is the 99% confidence interval for the mean sleep time (in hours) when the GA score is 6.2? (Carry your intermediate computations to at least four decimal places, and round your answer to at least two decimal places.) Lower limit: Upper limit: X

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter4: Equations Of Linear Functions
Section4.5: Correlation And Causation
Problem 15PPS
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A well-known psychologist has established what she calls her Generalized Anxiety (GA) scale. The GA scale, which is a scale from 0 to 10, measures the "general anxiety" of an individual, with higher GA scores corresponding to more anxiety. We'd like to make predictions about individuals' sleep behavior based on their GA scores. We've collected bivariate data that give the GA score (denoted by x) and the number of hours of sleep last night (denoted by y) for each of the 11 adults participating in a study. The least-squares regression equation for our data is y = 8.92-0.25x. We have used this equation to predict tonight's sleep time for a woman whose GA score is 6.2. We're now interested in both a prediction interval for her sleep time and a confidence interval for the mean sleep time of individuals with her GA score. We have computed the following for our data. • mean square error (MSE)≈ 0.601 (6.2-x)² ● 11 + 11 2 Σ (x₁ - x)² i=1 ≈ 0.1110, where x₁, x2, Lower limit: X11 Based on this information, and assuming that the regression assumptions hold, answer the questions below. (If necessary, consult a list of formulas.) Upper limit: denote the GA scores in the sample, and x denotes their mean (a) What is the 99% confidence interval for the mean sleep time (in hours) when the GA score is 6.2? (Carry your intermediate computations to at least four decimal places, and round your answer to at least two decimal places.) X
Transcribed Image Text:A well-known psychologist has established what she calls her Generalized Anxiety (GA) scale. The GA scale, which is a scale from 0 to 10, measures the "general anxiety" of an individual, with higher GA scores corresponding to more anxiety. We'd like to make predictions about individuals' sleep behavior based on their GA scores. We've collected bivariate data that give the GA score (denoted by x) and the number of hours of sleep last night (denoted by y) for each of the 11 adults participating in a study. The least-squares regression equation for our data is y = 8.92-0.25x. We have used this equation to predict tonight's sleep time for a woman whose GA score is 6.2. We're now interested in both a prediction interval for her sleep time and a confidence interval for the mean sleep time of individuals with her GA score. We have computed the following for our data. • mean square error (MSE)≈ 0.601 (6.2-x)² ● 11 + 11 2 Σ (x₁ - x)² i=1 ≈ 0.1110, where x₁, x2, Lower limit: X11 Based on this information, and assuming that the regression assumptions hold, answer the questions below. (If necessary, consult a list of formulas.) Upper limit: denote the GA scores in the sample, and x denotes their mean (a) What is the 99% confidence interval for the mean sleep time (in hours) when the GA score is 6.2? (Carry your intermediate computations to at least four decimal places, and round your answer to at least two decimal places.) X
(b) Consider (but do not actually compute) the 99% prediction interval for an individual value for sleep time when the GA score is 6.2. How would the confidence interval computed above compare to this prediction interval (assuming that both intervals are computed from the same sample data)? The confidence interval would be positioned to the left of the prediction interval. The confidence interval would be identical to the prediction interval. The confidence interval would have the same center as, but would be wider than, the prediction interval. The confidence interval would be positioned to the right of the prediction interval. The confidence interval would have the same center as, but would be narrower than, the prediction interval. (c) For the GA score values in this sample, 8.2 is more extreme than 6.2 is, that is, 8.2 is farther from the sample mean GA score than 6.2 is. How would the 99% confidence interval for the mean sleep time when the GA score is 8.2 compare to the 99% confidence interval for the mean sleep time when the GA score is 6.2? The intervals would be identical. The interval computed from a GA score of 8.2 would be wider but have the same center. The interval computed from a GA score of 8.2 would be wider and have a different center. The interval computed from a GA score of 8.2 would be narrower and have a different center. The interval computed from a GA score of 8.2 would be narrower but have the same center. X X Ś 5
Transcribed Image Text:(b) Consider (but do not actually compute) the 99% prediction interval for an individual value for sleep time when the GA score is 6.2. How would the confidence interval computed above compare to this prediction interval (assuming that both intervals are computed from the same sample data)? The confidence interval would be positioned to the left of the prediction interval. The confidence interval would be identical to the prediction interval. The confidence interval would have the same center as, but would be wider than, the prediction interval. The confidence interval would be positioned to the right of the prediction interval. The confidence interval would have the same center as, but would be narrower than, the prediction interval. (c) For the GA score values in this sample, 8.2 is more extreme than 6.2 is, that is, 8.2 is farther from the sample mean GA score than 6.2 is. How would the 99% confidence interval for the mean sleep time when the GA score is 8.2 compare to the 99% confidence interval for the mean sleep time when the GA score is 6.2? The intervals would be identical. The interval computed from a GA score of 8.2 would be wider but have the same center. The interval computed from a GA score of 8.2 would be wider and have a different center. The interval computed from a GA score of 8.2 would be narrower and have a different center. The interval computed from a GA score of 8.2 would be narrower but have the same center. X X Ś 5
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