II. Given Data

PROBLEM: An industrial engineer is investigating the effect of four assembly methods (A, B, C, D) on the assembly time for a color television component. Four operators are selected for the study. Furthermore, the engineer knows that each assembly method produces such fatigue that the time required for the last assembly may be greater than the time required for the first, regardless of method. That is, a trend develops in the required assembly time. To account for this source of variability, the engineer uses the Latin square design shown below. Analyze the data and draw appropriate conclusions.

Order of Assemby Operator_1 Operator_2 Operator_3 Operator_4
1 C = 10 D = 14 A = 7 B = 8
2 B = 7 C = 18 D = 11 A = 8
3 A = 5 B = 10 C = 11 D = 9
4 D = 10 A = 10 B = 12 C =14

III. Exploratory Data Analysis

Descriptive Statistic:

Method mean median sd
A 7.50 7.5 2.081666
B 9.25 9.0 2.217356
C 13.25 12.5 3.593976
D 11.00 10.5 2.160247

Interpretation: The result above shows descriptive statistics such as the mean, median, and standard deviation by assembly time. Among the four assembly methods, Method A and Method B have the lowest mean assembly times, with Method A having a slightly lower average time than Method B. Method C has the highest mean assembly time, significantly higher than Method C. Method D falls in between, with an average assembly time close to that of Method B but with higher variability. It’s important to consider factors such as variability (standard deviation) when assessing the performance of each method, as Methods C and D show more variability compared to A and B.

Graph (Interaction-plot):

Interpretation: The plot visualizes how assembly time varies across different assembly orders, with separate lines for each assembly method (A, B, C, and D). The y-axis represents the assembly time, and different operators are represented by different colors. We see that the different operators have varying impacts on assembly time for each assembly method and order of assembly combination. Operator 2 and Order 2 show more variability for the corresponding assembly method compared to others. Thus, there are interactions or differences between methods, orders, and operators in terms of their effect on assembly time.

IV. Experimental Question

Is there a significant difference between assembly methods regarding assembly time, at the significance level of α = 0.05?

V. Null and Alternative Hypothesis

Use hypothesis testing at the significance level of α = 0.05.

Null Hypothesis: There is no significant difference between assembly methods regarding assembly time.

Ho: μ1 = μ2 = μ3 = μ4 (This means that the mean assembly time in each assembly method is equal.)

Alternative Hypothesis: There is significant difference between assembly methods regarding assembly time.

Ha: μi ≠ μj (At least one of the means is different from the others.)

ANOVA TABLE

The model parameters:

Analysis of Variance Table

Response: Time
          Df Sum Sq Mean Sq F value   Pr(>F)   
Order      3   18.5  6.1667  3.5238 0.088519 . 
Operator   3   51.5 17.1667  9.8095 0.009926 **
Method     3   72.5 24.1667 13.8095 0.004213 **
Residuals  6   10.5  1.7500                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpretation:

The p-value associated with the operator variable is approximately 0.0099, and the method is approximately 0.0042, which is less than the typical significance level of 0.05. This indicates that the operator and method are both statistically significant in explaining the variation in time at the chosen significance level. While the order of assembly does not appear to be statistically significant as its p-value (0.0885) is greater than 0.05,The overall ANOVA F-test (F value) indicates that at least one of the predictor variables (order, operator, or method) has a significant effect on time because the overall p-value is less than 0.05. And since the method F-value of 13.81 implies the model is significant, using the α = 0.05 level of significance, the critical value of F_(0.05),(3,6) = 4.76, which is 13.81 > 4.76. Thus, we will reject the null hypothesis, and then we can accept the alternative hypothesis that there is a significant difference between assembly methods regarding assembly time.

VI. Syntax

Below is the code to compute some basic descriptive statistics for our dataset. But first, we used the read_xlsx() function to chunk our datasets, along with the kable() function to visualize our data clearly. Also, we used the data.frame() function to organize and store our data.

dats<-read_xlsx("E:/Experimental Design//Exercise 3.xlsx")
kable(dats, format = "html") %>%
  kable_styling(full_width = FALSE) %>%
  row_spec(0, bold = TRUE, color = "black", background = "skyblue") %>%
  row_spec(1:4, background = "white")
dat <- data.frame(
  Order = factor(c("1","2", "3","4","1","2", "3","4","1","2", "3","4","1","2", "3","4")),
  Operator = factor(c("1","1","1","1","2","2","2","2","3","3","3","3","4","4","4","4")),
  Method = factor(c("C","B","A","D","D","C","B","A","A","D","C","B","B","A","D","C")),
  Time = c(10,7,5,10,14,18,10,10,7,11,11,12,8,8,9,14))
dat
Summary<-dat%>%
  group_by(Method)%>%
  summarise(mean=mean(Time),median = median(Time),sd= sd(Time))

kable(Summary, format = "html") %>%
  kable_styling(full_width = FALSE) %>%
  row_spec(0, bold = TRUE, color = "black", background = "skyblue") %>%
  row_spec(1:4, background = "white")

Before actually performing the ANOVA in R, visualize the data in relation to the research question. This can be done with the interaction plot function in base R to see the relationships between the variables in our dataset.

ggplot(dat, aes(x = Order, y = Time, color =Operator  , group = Method )) +
  geom_line() +
  geom_point() +
  labs(x = "Order of Assembly",
    y = "Time",
    caption = "Figure 1.The effect of four assembly methods (A, B, C, D) on the assembly time"
  ) +
  theme_minimal()

Now, only the ANOVA test can help us answer the initial research question. The provided code conducts a linear regression analysis and performs an analysis of variance (ANOVA) to examine the relationships between the variables in our dataset. We used some factor conversion, such as the as.factor() function, to represent categorical variables in statistical analysis. Also, the anova(model) function is used to perform an analysis of variance (ANOVA) on the fitted linear regression model, and then we have created a linear regression model, or lm() function, with the following formula:

Method = as.factor(dat$Method)
Order = as.factor(dat$Order)
Operator = as.factor(dat$Operator)
model <- lm(formula = Time ~ Order + Operator + Method, data = dat)
anova(model)