Question

3.1. A researcher is interested in determining whether a new and cheaper method of analysis to determine the fat content of small organisms works as well as the older method used for many years. She has decided to split 15 samples and determine fat content in the two halves using the two methods. She has decided that simple linear regression will be the best method of analysis because she can test the hypothesis that the two methods give a 1:1 relationship (slope = 1) over the working range of the methods. The data collected are (mg fat/g tissue):

Sample#	Old Method (mg/g)	New Method (mg/g)
1	18.6	18.1
2	5.9	6.1
3	19.1	19.6
4	23.2	20.1
5	12.4	8.1
6	14.2	15.2
7	18.3	15.6
8	15.6	13.5
9	14.8	10.9
10	27.6	27.8
11	13.6	12.1
12	17.5	19.2
13	26.4	24.6
14	6.4	4.5
15	9.2	8.1

A). Create a well labelled xy scatter plot (where x = old method) on the spreadsheet page with the data; make note of the general pattern of the plotted data and whether there are any suspect data points.

B). Use the advanced regression analysis tool to conduct a complete simple linear regression analysis for the data. You will not require the options for "residuals" for this analysis. Remember to complete the seven steps of hypothesis testing in the spaces provided on the worksheet.

C). Use the linear regression equation determined in part "B" to calculate a set of "predicted y values" for each observed x value.

D). What are the predicted y values (ie predicted “fat content using the new method”) for the following x values? a). 11.3 mg/g b). 17.2 mg/g c). 21.5 mg/g d). 148.4 mg/g

E). Whether the regression from question part “B” is significant or not, use the t-test for slope to test the hypothesis that the slope = 1.0. Remember to complete the five steps of hypothesis testing in the spaces provided on the worksheet. What does this analysis tell you about the two methods?

**please show how you got the answers so that I can learn and understand, as well as each of the steps for the seven steps of hypothesis testing so that I can see each step clearly **

Answer 1

Ans a ) the scatterplot of the data is

there is a positive linear relationship between the fat content using the new method and old method . there is no any suspect data points.

B ) using excel>data>data analysis>Regresion

we have

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.964485
R Square	0.930232
Adjusted R Square	0.924866
Standard Error	1.857378
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	1	597.9719	597.9719	173.3325	6.82E-09
Residual	13	44.8481	3.449854
Total	14	642.82
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-1.50844	1.335397	-1.12958	0.279067	-4.39339	1.376516
Old Method (mg/g)	1.013701	0.076996	13.16558	6.82E-09	0.84736	1.180041

B). Use the advanced regression analysis tool to conduct a complete simple linear regression analysis for the data.

the null and alternative hypothesis is

Ho: there is not a significant relationship between the old and new method

Ha: there is significant relationship between the old and new method

the value of F stat = 173.333

p value = 0.0000

since p value is less than 0.05 so we reject Ho and conclude that there is significant relationship between the old and new method .

C). Use the linear regression equation determined in part "B" to calculate a set of "predicted y values" for each observed x value. (2 marks)

Old Method (mg/g)	New Method (mg/g)	predicted
18.6	18.1	17.3464
5.9	6.1	4.472396
19.1	19.6	17.85325
23.2	20.1	22.00942
12.4	8.1	11.06145
14.2	15.2	12.88611
18.3	15.6	17.04229
15.6	13.5	14.3053
14.8	10.9	13.49433
27.6	27.8	26.46971
13.6	12.1	12.27789
17.5	19.2	16.23133
26.4	24.6	25.25327
6.4	4.5	4.979246
9.2	8.1	7.817609

D). What are the predicted y values (ie predicted “fat content using the new method”) for the following x values?

(4 marks) a). 11.3 mg/g b). 17.2 mg/g c). 21.5 mg/g d). 148.4 mg/g

	value	predicted
a	11.3	9.946381
b	17.2	15.92722
c	21.5	20.28613
d	148.4	148.9248

E). Whether the regression from question part “B” is significant or not,

the regression is significant as we shown in part B

use the t-test for slope to test the hypothesis that the slope = 1.0.

the null and alternative hypothesis is

Ho:b1=1

Ha:b1 1

t = = 0.178

t at 95% with 13 df = 2.179

since t < 2.179 so we do not reject ho and conclude that the slope = 1.0

It tells us that there is significant relationship between the old and new method and about 93.02% variation in new method can be explained by the old method through the regression line equation.