Question

Twenty-one mature flowers of a particular species were dissected, and the number of stamens and carpels present in each flower were counted. x, Stamens 52 68 70 38 61 51 56 65 43 37 36 74 38 35 45 72 59 60 73 76 68 y, Carpels 21 30 29 19 18 28 31 31 18 24 21 28 29 24 26 22 34 28 32 36 33 (a) Is there sufficient evidence to claim a linear relationship between these two variables at α = .05? (i) Find r. (Give your answer correct to three decimal places.) (iii) State the appropriate conclusion. Reject the null hypothesis, there is not significant evidence to claim a linear relationship. Reject the null hypothesis, there is significant evidence to claim a linear relationship. Fail to reject the null hypothesis, there is significant evidence to claim a linear relationship. Fail to reject the null hypothesis, there is not significant evidence to claim a linear relationship. (b) What is the relationship between the number of stamens and the number of carpels in this variety of flower?. (Give your answers correct to two decimal places.) = + x (c) Is the slope of the regression line significant at α = .05? (i) Find t. (Give your answer correct to two decimal places.) (ii) Find the P-value. (Give your answer bounds exactly.) < p < (iii) State the appropriate conclusion. Reject the null hypothesis, there is not evidence of a significant slope. Reject the null hypothesis, there is evidence of a significant slope. Fail to reject the null hypothesis, there is evidence of a significant slope. Fail to reject the null hypothesis, there is not evidence of a significant slope. (d) Find the 95% prediction interval for the number of carpels that one would expect to find in a mature flower of this variety if the number of stamens were 65. (Give your answers correct to one decimal place.) Lower Limit Upper Limit

Answer 1

a)

Coeffiecient of correlation r = 0.5519

Hypothesis:

H0​: ρ=0

HA​: ρ̸​=0

df = n-2 = 21 - 2 = 19

α = 0.05

t stat = r * SQRT((n-2)/(1-r^2)) = 0.5519 * SQRT((19/(1-0.5519^2)) = 2.885

P value = 0.0095 (Use t table)

P value < 0.05, Reject H0

Reject the null hypothesis, there is significant evidence to claim a linear relationship

b)

Y = 15.0388 + 0.2092 * X

c)

slope of the regression line significant at α = .05

Hypothesis:

H0: β1 = 0

Ha: β1 not = 0

t stat = Slope / Sb1 = 0.2092/0.0725 = 2.885

P value = 0.0095

0.005 < P value < 0.01

P value < 0.05, It is significant

Reject the null hypothesis, there is evidence of a significant slope

d)

σ^2 = MSE

(Use t table for critical and p values)