Exercise 2

1. An agricultural engineer obtained the following data on two methods of drying corned and asked for a statistical analysis:

Drying rate
With preheater	Without preheater
16	20
12	10
22	21
14	10
19	12

a) What questions would you ask him?

b) Under what circumstances would you be justified in analyzing the data using (i) a paired t test (ii) an unpaired t test, (iii) something else?

c) Analyze the data for options (i) and (ii)

2. The following are results from a larger study on the effects of nalbuphine. The measured response is the change in pupil diameter (in millimeters) after a dose of nalbuphine (B) and morphine (A). Assume that the 11 subjects were randomly allocated to the drugs.

A	+2.4	+0.08	+0.8	+2.0	+1.9	+1.0
B	+0.4	+0.2	-0.3	+0.8	0.0

What is a 95% confidence interval for the difference in the population means? What is the 90% confidence interval? State your assumptions.

[Source: These data are from H.W.Elliot et al J. Med. 1, 77(1970)]

3. A chemist performed the following experiments, randomizing the order of the runs within each week. Analyze the results.

Variable Yield

run	temperature	catalyst	pH	week1	week2
1	-	-	-	60.4	62.1
2	+	-	-	75.4	73.1
3	-	+	-	61.2	59.6
4	+	+	-	67.3	66.7
5	-	-	+	66.0	63.3
6	+	-	+	82.9	82.4
7	-	+	+	68.1	71.3
8	+	+	+	75.3	77.1

The minus and plus values of the three variables are as follows:

Temperature

(deg C)

Catalyst

(%)

130

6.8

150

6.9

The object was to obtain increased yield. Write a report for the chemist , asking appropriate questions and giving advice on how the investigation might proceed .

4. A 2³ factorial design has been run in duplicate for the purpose of studying in a research laboratory the effects of three variables ( temperature, concentration, and time ) on the amount of a certian metal recovered from an ore sample of given weight.

Analyze these data, and state the practical conclusions you reach.

Temperature	Concentration	Time	Response:weight recovered	of metal (grams)
-	-	-	80	62
+	-	-	65	63
-	+	-	69	73
+	+	-	74	80
-	-	+	81	79
+	-	+	84	86
-	+	+	91	93
+	+	+	93	93

Code:

- 1600 degC 30% 1 hour

+ 1900 degC 35% 3 hours

i. Do you think that these data contain a discrepant value ?

ii. Perform an analysis, using all the data, and plot residuals.

iii. Perform an approximate analysis, omitting the discrepant value.

iv. Report your findings.

5. A 2³factorial design on radar tracking was replicated 20 times with these results :

1	2	3	Ybar	S
-	-	-	7.76	0.53
+	-	-	10.13	0.74
-	+	-	5.86	0.47
+	+	-	8.76	1.24
-	-	+	9.03	1.12
+	-	+	14.59	3.22
-	+	+	9.18	1.80
+	+	+	13.04	2.58

Where Ybar is the average acquisition time ( in seconds ) , and S is the calculated standard deviation of Ybar. The variables were:

Modified ( - ) or standard ( + ) mode of operation .

1. Short ( - ) or long ( + ) range.

2. Slow ( - ) or fast ( + ) plane.

Analyze the results. Do you think a transformation of the data should be made ?

6. Some tests were carried out on a newly designed carburator. Four variables were studied as follows :

Variable		-	+
A	tension on spring	low	high
B	air gap	narrow	open
C	size of aperture	small	large
D	rate of flow of gas	slow	rapid

The immediate object was to find the effects of these changes on the amount of unburned hydrocarbones in the engine exhaust gas. The following results were obtained:

A	B	C	D	Unburned hydrocarbons
-	+	+	+	8.2
-	-	+	-	1.7
-	-	-	+	6.2
+	-	-	-	3.0
+	-	+	+	6.8
+	+	+	-	5.0
-	+	-	-	3.8
+	+	-	+	9.3

Stating any assumptions, analyze these data. It was hoped to develop a design giving lower levels of unburned hydrocarbons. Write a report, asking appropriate questions and making tentative suggestions for further work.