Question

Consider the data.

xi	2	6	9	13	20
yi	7	17	10	28	24

(a)

What is the value of the standard error of the estimate? (Round your answer to three decimal places.)

_________

(b)

Test for a significant relationship by using the t test. Use α = 0.05.

State the null and alternative hypotheses.

H₀: β₀ = 0
H_a: β₀ ≠ 0

H₀: β₁ = 0
H_a: β₁ ≠ 0    

H₀: β₀ ≠ 0
H_a: β₀ = 0

H₀: β₁ ≠ 0
H_a: β₁ = 0

H₀: β₁ ≥ 0
H_a: β₁ < 0

Find the value of the test statistic. (Round your answer to three decimal places.)

_______

Find the p-value. (Round your answer to four decimal places.)

p-value = _______

State your conclusion.

Reject H₀. We conclude that the relationship between x and y is significant.

Do not reject H₀. We conclude that the relationship between x and y is significant.     

Do not reject H₀. We cannot conclude that the relationship between x and y is significant.

Reject H₀. We cannot conclude that the relationship between x and y is significant.

(c)

Use the F test to test for a significant relationship. Use α = 0.05.

State the null and alternative hypotheses.

H₀: β₀ = 0
H_a: β₀ ≠ 0

H₀: β₁ = 0
H_a: β₁ ≠ 0    

H₀: β₁ ≠ 0
H_a: β₁ = 0

H₀: β₁ ≥ 0
H_a: β₁ < 0

H₀: β₀ ≠ 0
H_a: β₀ = 0

Find the value of the test statistic. (Round your answer to two decimal places.)

_______

Find the p-value. (Round your answer to three decimal places.)

p-value = _______

What is your conclusion?

Do not reject H₀. We cannot conclude that the relationship between x and y is significant.

Reject H₀. We cannot conclude that the relationship between x and y is significant.     

Do not reject H₀. We conclude that the relationship between x and y is significant.

Reject H₀. We conclude that the relationship between x and y is significant.

Answer 1

We have used R. On right side is code and on left side is output.

a) The value of the standard error of the estimate is  5.5842.

b) Null hypothesis will be

H0: β0 = 0

Alternative hypothesis will be

Ha: β1 ≠ 0

t-value= 1.289 p-value= 0.288

As p-value>0.05, we fail to reject the null hypothesis.

Do not reject H0. We cannot conclude that the relationship between x and y is significant.

c) Using F-test, we have

H0: β0 = 0
Ha: β0 ≠ 0

F statistic is defined as

So, we have

The value of the test statistic is 4.4255 and p-value is 0.1261.  Since p-value ≥ α (0.05), we accept the H0.

Do not reject H0. We conclude that the relationship between x and y is significant.