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	19	32	27	21	18	30	31	31	20	26	23	28	27	24	28	22	34	26	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 98% 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 45. (Give your answers correct to one decimal place.)

Lower Limit
Upper Limit

Answer 1

Using Excel, go to Data, select Data Analysis, choose Regression at confidence level 98%.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.479
R Square	0.229
Adjusted R Square	0.189
Standard Error	4.685
Observations	21
ANOVA
	df	SS	MS	F	Significance F
Regression	1	123.995	123.995	5.650	0.028
Residual	19	416.958	21.945
Total	20	540.952
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 98.0%	Upper 98.0%
Intercept	17.259	4.243	4.067	0.001	8.378	26.140	6.484	28.034
Stamens	0.175	0.073	2.377	0.028	0.021	0.328	-0.012	0.361

a) H0: There is no linear relationship between x and y

H1: There is a linear relationship between x and y

p-value (Significance F) = 0.028

Since p-value is less than 0.05, we reject the null hypothesis and conclude that there is a linear relationship between x and y.

i) r (R Square) = 0.229

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

b) Since coefficient of stamens is positive, there exists a + relationship between number of stamens and the number of carpels.

c) H0: β1 = 0, slope of the regression line is not significant

H1: β1 ≠ 0, slope of the regression line is significant

i) t-stat = 2.38

ii) p-value = 0.03

iii) Since p-value is less than 0.05, we reject the null hypothesis and conclude that slope of the regression line is significant.

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

d) 98% prediction interval if the number of stamens were 45:

y = 17.259 + 0.175*x = 17.259 + 0.175*45 = 25.118

α =1-0.98 = 0.02

t1−α/2,n−2 = t1−0.02/2,21−2 (Using Excel function T.INV.2T(probability,df)) = T.INV.2T(0.02,19) = 2.539

SE = 4.685

Lower Limit = y - t*SE = 25.118-0.02*4.685 = 25.0

Upper Limit = y + t*SE = 25.118+0.02*4.685 = 25.2