To test whether the mean drying time is the same for each brand of robot, we conduct a one-way ANOVA (Analysis of Variance). Here's the detailed breakdown:

Step 1: Organize the Data

The drying times for the four robots are:

• Robot 1: 128, 137, 135, 140, 141
• Robot 2: 144, 133, 135, 141, 130
• Robot 3: 133, 143, 124, 130, 131
• Robot 4: 150, 142, 146, 136, 153

Step 2: Formulas Used in ANOVA

Total Sum of Squares (SS$_T$):

$SS_T = \sum_{i=1}^k \sum_{j=1}^n (X_{ij} - \bar{X})^2$

Treatment Sum of Squares (SS$_{TR}$):

$SS_{TR} = n \sum_{i=1}^k (\bar{X}_i - \bar{X})^2$

Error Sum of Squares (SS$_E$):

$SS_E = SS_T - SS_{TR}$

Mean Squares:

• $MS_{TR} = \frac{SS_{TR}}{k-1}$
• $MS_E = \frac{SS_E}{N-k}$

Where:

• $k$: Number of treatments (robots).
• $n$: Number of observations per treatment.
• $N$: Total number of observations ($k \times n$).

Test Statistic:

$F = \frac{MS_{TR}}{MS_E}$

Step 3: Compute Values

Compute Means

• Grand mean ($\bar{X}$): Mean of all observations.
• Treatment means ($\bar{X}_i$): Mean for each robot.

Data Totals:

• $n = 5$ (observations per robot),
• $k = 4$ (robots),
• $N = 20$ (total observations).

I will calculate these values step np import pandas as pd

Data for the drying times of robots

data = { "Robot 1": [128, 137, 135, 140, 141], "Robot 2": [144, 133, 135, 141, 130], "Robot 3": [133, 143, 124, 130, 131], "Robot 4": [150, 142, 146, 136, 153], }

Convert to DataFrame for analysis

df = pd.DataFrame(data)

Number of groups (robots) and observations per group

k = len(df.columns) # Number of robots n = len(df) # Observations per robot N = k * n # Total number of observations

Calculate means

grand_mean = df.values.flatten().mean() group_means = df.mean()

Total Sum of Squares (SST)

SST = np.sum((df.values.flatten() - grand_mean) ** 2)

Treatment Sum of Squares (SSTR)

SSTR = n * np.sum((group_means - grand_mean) ** 2)

Error Sum of Squares (SSE)

SSE = SST - SSTR

Mean Squares

MSTR = SSTR / (k - 1) MSE = SSE / (N - k)

F-Statistic

F_statistic = MSTR / MSE

Results

results = { "Sum of Squares (Treatment)": SSTR, "Sum of Squares (Error)": SSE, "Mean Squares (Treatment)": MSTR, "Mean Squares (Error)": MSE, "F-Statistic": F_statisticHere are the computed values for the one-way ANOVA:

1. Sum of Squares (Treatment): $464.80$
2. Sum of Squares (Error): $610.00$
3. Mean Squares (Treatment): $154.93$
4. Mean Squares (Error): $38.12$
5. F-Statistic: $4.06$

Next Steps:

To determine if the differences in means are statistically significant, compare the calculated $F$-statistic to the critical $F$-value for $(k-1, N-k)$ degrees of freedom at the specified significance level.

Would you like me to proceed with this comparison or provide further clarification? Here are five related questions you might consider:

1. What is the critical $F$-value for this test at a 0.05 significance level?
2. Can we calculate the p-value for the $F$-statistic?
3. How does one interpret the results of the ANOVA test?
4. If the null hypothesis is rejected, what follow-up test should we perform to identify differences between groups?
5. How do changes in sample size affect the power of the ANOVA test?

Tip: ANOVA tests are powerful but assume normality and equal variance across groups. Always verify these assumptions