PROMBLEM 1
Three different detergent makers (A, B, & C) all claim that their detergent gets clothes their "whitest white" in all water temps. To check the validity of theses claims, a consumer protection service called Consumers-R-Us tests the detergents using cold, warm, and hot water in standard washing machine. Consumer R Us purchased 36 identical white sheets and randomized 4 sheets to each detergent-water temp. Combination. The sheets were washed and then measured for their "whiteness score" as determined by laser equipment. The same automatic washing machine was used throughout.
DETERGENT / WATER TEMP / WHITENESS SCORESA / COLD / 17 28 22 21
A / WARM / 26 28 30 31
A / HOT / 47 57 52 54
B / COLD / 22 20 16 22
B / WARM / 42 40 48 36
B / HOT / 44 42 36 35
C / COLD / 27 33 20 18
C / WARM / 38 32 42 31
C / HOT / 48 44 56 54
1) Before carrying out any analysis, check the validity of any assumptions that are necessary for ANOVA tests. Write a brief statement as to why or why not assumptions are met/not met (specific)
· Since the p-value is greater than 0.05 we fail to reject the null hypothesis of equal variances. So there is not a significant difference between the variances.
2) Analyze the data with ANOVA. Check for detergent*temperature interactions and carry out the appropriate tests depending on your findings. Test the null hypothesis that the 3 detergents have the same mean whiteness score. Carry out appropriate post-hoc tests if the overall ANOVA test is significant at 0.05 (what tests carried out)?
· Since the p-value of the variable Water Temperature is less than 0.05 we conclude that Water Temperature shows significant source of variability. So the whiteness score averages of Hot, Warm and Cold water are different.
· Whiteness score is independent of the detergent used at this level of significance. We don’t have sufficient evidence to conclude that the whiteness score averages of detergents A, B and C are different.
· There is significant interaction between the variables Detergent and Water Temperature. So the effect of using different detergents is different at different temperatures.
3) What conclusions can you draw about the effects of detergent and water temp on the whiteness of the sheets?
· Detergents A, B, and C have different effects at different temperatures, but they are actually not different at whitening.
· Water temperature is an important factor on whiteness score.
PROBLEM TWO
Ajax products, inc., a start up manufacturing company, has a total of 4 machines (A, B, C, D) in its manufacturing plant that make product to sell. Each machine runs in 3 shifts per day and so there are 3 operators that run each machine (day, swing, and graveyard shifts). Operators only work on the machine that they are assigned to. There is concern lately about how much usable product is being manufactured, so data were collected on the number of usable product is being manufactured, so data were collected on the number of usable items manufactured during 2 recent days.
MACHINE / OPERATOR / # OF USABLE ITEMSA / 1 / 26 25
A / 2 / 25 26
A / 3 / 26 27
B / 1 / 26 27
B / 2 / 30 31
B / 3 / 24 26
C / 1 / 25 26
C / 2 / 25 27
C / 3 / 27 25
D / 1 / 26 24
D / 2 / 29 28
D / 3 / 26 28
1) Before carrying out any analysis, check the validity of any assumptions that are necessary for ANOVA tests. Write a brief statement as to why or why not assumptions are met/not met (specific)
· Since the p-value is greater than 0.05 we fail to reject the null hypothesis of equal variances. So there is not a significant difference between the variances.
2) Analyze the data with ANOVA. Carry out all hypothesis tests at 0.05. Do the machines or the operators or both represent a significant source of variability that the company should be concerned about? Please briefly explain
· Since the p-value of the variable operator is less than 0.05 we may conclude that operators show significant source of variability that the company should be concerned about.
· Number of usable items produced is independent of the machine used at this level of significance.
3) Briefly state your conclusions. What can AJAX do to improve their situation?
· Operator 1 is not as successful as operator 3 at producing usable items. They should train their operators and increase the efficiency of production.
PROBLEM 3
A study was conducted investigation whether emotions (under hypnosis) have the same effect on skin potential (measured in millivolts). Eight subjects were asked to display fear, joy, sadness, and calmness under hypnosis. The skin potential data for these four emotions are given in this table.
SUBJECT / FEAR / JOY / SADNESS / CALMNESS1 / 23.1 / 22.7 / 22.5 / 22.6
2 / 57.6 / 53.2 / 53.7 / 53.1
3 / 10.9 / 9.7 / 10.8 / 8.3
4 / 23.6 / 19.6 / 21.1 / 21.6
5 / 13.9 / 13.8 / 13.7 / 13.3
6 / 54.6 / 47.1 / 39.2 / 37.0
7 / 21.0 / 13.6 / 13.7 / 14.8
8 / 20.3 / 23.6 / 16.3 / 14.8
1) Before carrying out any analysis, check the validity of any assumptions that are necessary for ANOVA tests. Write a brief statement as to why or why not assumptions are met/not met (specific)
· Since the sig. probability is less than 0.05 we can not assume the variances between the four emotions are equal.
2) Analyze the data with ANOVA. Carry out an appropriate analysis to test whether or not the four emotions have the same skin potential. If the overall test to compare emotions is significant at 0.05, compare emotions pair wise to determine which comparisons are significant. Explain findings
· Since the p-value of the variable Emotion is less than 0.05 we conclude that Emotion shows significant source of variability. So the potential averages of Fear, Joy, Sadness, and Calmness are different.
· According to pairwise comparisons, Fear gives significantly different potential values than Calmness.
3) Briefly state your conclusions
· Emotions (under hypnosis) don’t have the same effect on skin potential (measured in millivolts). Fear has significantly different potential value than calmness.
PROBLEM 4
Acme pharmaceutical, inc., a leader in dermatology products, has manufacturing plants around the would to keep up with demand. Recently, the production of the number of tubes of skin cream has been less than expected. Therefore, 3 plants and 4 weeks during the year were randomly selected for a study of production from the many plants and weeks available. For each chosen week, the number of tubes made on two days during the week was recorded.
PLANT / WEEK / # OF TUBES PRODUCED1 / 1 / 216 204
1 / 2 / 205 206
1 / 3 / 203 204
1 / 4 / 202 203
2 / 1 / 205 204
2 / 2 / 205 206
2 / 3 / 206 208
2 / 4 / 209 210
3 / 1 / 211 213
3 / 2 / 211 212
3 / 3 / 213 215
3 / 4 / 214 216
1) Before carrying out any analysis, check the validity of any assumptions that are necessary for ANOVA tests. Write a brief statement as to why or why not assumptions are met/not met (specific)
· Since the p-value is greater than 0.05 we fail to reject the null hypothesis of equal variances. So there is not a significant difference between the variances.
2) Analyze the data with ANOVA. Carry out all hypothesis tests at 0.05 sig. please provide an ANOVA table and include the expected mean squares
· Since the p-value of the variable Plant is less than 0.05 we conclude that Plant shows significant source of variability. So the # of tubes produced is different for plants 1, 2, and 3.
· Week doesn’t have a significant effect on production at 5% significance level.
2) What conclusions can you draw about the effect of plant and week on the variability of the number of tubes produced?
· # of tubes produced significantly depends on the plant but not on the week of production.