Question

Researchers examined the relationship between Age (x) in years and blood-brain barrier (BBB) permeability (y) in transfer units. They collected a simple random sample of 24 adults from the New York City region and measured BBB permeability in the hippocampus. They decide to use least-squares regression to estimate the population regression line:

μy=α+βx

The results from their analysis include:

a=0.67,b=0.016,s=0.0181,SEa=0.84,SEb=0.0037,r2=0.84

(a) What does the estimate of the slope for the least-squares regression line tells us?

A. For every 1 year increase in Age, BBB permeability increases by 0.016 units, on average.

B. For every 1 year increase in Age, BBB permeability increases by 0.67 units, on average.

C. For every 1 unit increase in BBB permeability, Age increases by 0.016 years, on average.

D. For every 1 unit increase in BBB permeability, Age increases by 0.67 years, on average.

(b) We want to test whether or not there is a statistically significant linear relationship between x and y. What are the corresponding null and alternative hypotheses in scientific notation?

A. H0:β=0 and H1:β≠0
B. H0:b=0 and H1:b>0
C. H0:b=1 and H1:b≠1
D. H0:β=1 and H1:β≠1

c) What is the value of the test statistic for the test?

A. 0.7976

B. 0.884

C. 0.84

D. 4.3243

(d) What is the distribtion of this test statistic?

A. Normal(0,1)
B. t(df=23)
C. Chi-square(df=23)
D. t(df=22)

(e) What is the P-value of this test?

A. Greater than 0.10

B. Between 0.05 and 0.10

C. Between 0.01 and 0.05

D. Less than 0.01

(f) What is correlation coefficient between BBB permeability and Age?

A. -0.9165

B. 0.84

C. 0.9165

D. -0.884

(g) What is a 95% confidence interval for the slope in the simple linear regression model?

A. ( 0.0087 , 0.0233 )

B. ( 0.0083 , 0.0237 )

C. ( -0.0195 , 0.0515 )

D. ( -1.0722 , 2.4122 )

(h) Which of the following is the critical value ( t∗ or z∗ ) used to construct the 95% confidence interval?

A. 1.645

B. 1.717

C. 1.96

D. 2.074

(i) The correct interpretation for this confidence interval is that

A. We are 95% confident that BBB permeability in the hippocampus changes, on average, by some amount in this interval for every additional year that adults age.

B. We are 95% confident that BBB permeability in the hippocampus changes by some amount in this interval for every additional year that adults age.

C. For 95% of adults, a one year increase in age results in a change of BBB permeability in the hippocampus by some amount in this interval.

D. None of the interpretations are correct.

Answer 1

a)

A. For every 1 year increase in Age, BBB permeability increases by 0.016 units, on average.

b)

A. H0:β=0 and H1:β≠0

c)

test statistic=0.016/0.0037 =4.3243

d)

t(df=22)

e)

D. Less than 0.01

f)

Correlation =sqrt(0.84)=0.9165

g)

for 95 % confidence and 22degree of freedom critical t=					2.0740
95% confidence interval =b1 -/+ t*standard error=					(0.0083,0.0237)

h)

2.074

i)

A. We are 95% confident that BBB permeability in the hippocampus changes, on average, by some amount in this interval for every additional year that adults age.