Question

An employee at a coffee shop hypothesizes that the harder the espresso grounds are tamped down into the portafilter before​ brewing, the longer the separation time of the​ heart, body, and crema will be. The accompanying data table shows the results of this experiment. The independent variable tamp measures the​distance, in​ inches, between the espresso grounds and the top of the portafilter. The dependent variable time is the number of seconds the​ heart, body, and crema are separated. Complete parts​ (a) through​ (f) below.

Shot   Tamp   Time
1   0.20   15
2   0.55   15
3   0.25   12
4   0.15   13
5   0.20   15
6   0.40   14
7   0.25   15
8   0.50   9
9   0.15   17
10   0.30   13
11   0.20   10
12   0.15   15
13   0.40   18
14   0.45   19
15   0.15   15

Part a) Use the​ least-squares method to develop a single regression equation with Time as the dependent variable and Tamp as the independent variable. (Can you show me the step by step process using PHSTAT on all answers)

part b) Predict the mean separation time for a tamp distance of 0.45 inch.

part c) Plot the residuals versus the time order of experimentation. Are there any noticeable​ patterns?

part d) Compute the​ Durbin-Watson statistic. At the 0.05 level of​ significance, is there evidence of positive autocorrelation among the​ residuals?

Part E) Based on the results of​ (c) and​ (d), is there reason to question the validity of the​ model?

Answer 1

We will solve this problem with the help of Excel.

Part a) Use the​ least-squares method to develop a single regression equation with Time as the dependent variable and Tamp as the independent variable. (Can you show me the step by step process using PHSTAT on all answers)

Load the data into Excel.

Go to Data>Megastat.

Select the option Correlation/Regression and go to Regression.

Select Tamp as the independent variable, x.

Select Time as the dependent variable, y.

Click OK.

The output will be as follows:

	r²	0.001	n	15
	r	0.022	k	1
	Std. Error	2.791	Dep. Var.	Time
ANOVA table
Source	SS	df	MS	F	p-value
Regression	0.0509	1	0.0509	0.01	.9368
Residual	101.2824	13	7.7910
Total	101.3333	14
Regression output					confidence interval
variables	coefficients	std. error	t (df=13)	p-value	95% lower	95% upper
Intercept	14.2082
Tamp	0.4364	5.3984	0.081	.9368	-11.2262	12.0990

We have the simple linear regression equation from the output:
y = 14.2082 + 0.4364*x

Or

Time = 14.2082 + 0.4364*Tamp

part b) Predict the mean separation time for a tamp distance of 0.45 inch.

We are given Tamp = 0.45 inch

x = 0.45 inch

Putting this value into our regression equation, we get:

y = 14.2082 + 0.4364*0.45

y = 14.405

Therefore, the mean separation time for a tamp distance of 0.45 inch is 14.405.

part c) Plot the residuals versus the time order of experimentation. Are there any noticeable​ patterns?

Load the data into Excel.

Go to Data>Megastat.

Select the option Correlation/Regression and go to Regression.

Select Tamp as the independent variable, x.

Select Time as the dependent variable, y.

Select Plot residuals by observation.

Click OK.

The output will be as follows:

The residual plot shows a fairly random pattern - the first and second residual is positive, the next two are negative, the fifth is positive, and the last residual is also positive. This random pattern indicates that a linear model provides a decent fit to the data.

part d) Compute the​ Durbin-Watson statistic. At the 0.05 level of​ significance, is there evidence of positive autocorrelation among the​ residuals?

Load the data into Excel.

Go to Data>Megastat.

Select the option Correlation/Regression and go to Regression.

Select Tamp as the independent variable, x.

Select Time as the dependent variable, y.

Select Durbin-Watson.

Click OK.

The output will be as follows:

Observation	Time	Predicted	Residual
1	15.0	14.3	0.7
2	15.0	14.4	0.6
3	12.0	14.3	-2.3
4	13.0	14.3	-1.3
5	15.0	14.3	0.7
6	14.0	14.4	-0.4
7	15.0	14.3	0.7
8	9.0	14.4	-5.4
9	17.0	14.3	2.7
10	13.0	14.3	-1.3
11	10.0	14.3	-4.3
12	15.0	14.3	0.7
13	18.0	14.4	3.6
14	19.0	14.4	4.6
15	15.0	14.3	0.7
	Durbin-Watson =	1.92

The​ Durbin-Watson statistic from the output is 1.92.

Yes, there is evidence of positive autocorrelation among the​ residuals. In positive autocorrelation, consecutive errors usually have the same sign: positive residuals are almost always followed by positive residuals, while negative residuals are almost always followed by negative residuals. The first and the second residual is positive, the next two are negative, the fifth is positive, and the last residual is also positive.

Part E) Based on the results of​ (c) and​ (d), is there reason to question the validity of the​ model?

Mutual information will very simply tell us if variable X is related to variable Y, and how much uncertainty is reduced in predicting Y if the uncertainty in knowing X is quantified. Here, variable X is related to variable Y positively having a linear relation and thus, there is no reason to question the validity of the​ model.