Question 1204266: Once a baby is born, an option becomes available to harvest the stem cells of the child from the umbilical cord. This is done at the hospital, in the labour ward. The specimen must be cooled very quickly and maintained at the same temperature for extended periods (often many years). Three competing refrigeration units are tested against one another by the claim that they are able to reduce the temperature of the specimen, in the hospital, to the required level in a minimum amount of time. The refrigeration units are tested at 27 randomly allocated labour wards around the Western Cape. The rate at which the desired temperature is reached (in seconds) is below.
Refrigeration Unit : Temperature rate (in seconds)
Unit 1 : 82; 61; 63; 91; 88; 71; 59; 65; 58; 49
Unit 2 : 67; 58; 72; 71; 54; 79; 44; 60; 77
Unit 3 : 91; 112; 84; 81; 76; 54; 82; 64
Calculate at a 10% level of significance whether there is a significant difference between the mean time until the correct temperature is reached in the three refrigeration units.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website!
**Interpretation:**
* **ANOVA Results:**
* The ANOVA table will provide the F-statistic, degrees of freedom, and p-value.
* If the p-value is less than 0.10 (our significance level), we reject the null hypothesis and conclude that there is a significant difference in the mean time until the correct temperature is reached among the three refrigeration units.
* **Boxplot:**
* The boxplot will visually represent the distribution of temperature rates for each unit, helping to identify any significant differences in central tendency and variability.
**Note:**
* This analysis assumes that the data within each group (refrigeration unit) is normally distributed and that the variances between groups are equal. You may need to perform additional tests (e.g., Shapiro-Wilk test for normality, Levene's test for homogeneity of variances) to assess these assumptions.
**Conclusion:**
By performing the ANOVA and examining the boxplot, you can determine whether there is a significant difference in the mean time until the correct temperature is reached among the three refrigeration units at the 10% level of significance.
```R
# Sample data
unit_1 <- c(82, 61, 63, 91, 88, 71, 59, 65, 58, 49)
unit_2 <- c(67, 58, 72, 71, 54, 79, 44, 60, 77)
unit_3 <- c(91, 112, 84, 81, 76, 54, 82, 64)
# Combine data and create a factor variable for units
data <- data.frame(
time = c(unit_1, unit_2, unit_3),
unit = factor(rep(1:3, c(length(unit_1), length(unit_2), length(unit_3))))
)
# Perform one-way ANOVA
model <- aov(time ~ unit, data = data)
# Summary of the ANOVA model
summary(model)
# Create a boxplot to visualize the data
boxplot(time ~ unit, data = data,
main = "Boxplot of Temperature Rates",
xlab = "Refrigeration Unit",
ylab = "Time (seconds)")
```
**Interpretation:**
1. **ANOVA Results:**
* The ANOVA table will provide the F-statistic and its associated p-value.
* If the p-value is less than 0.10 (our significance level), we reject the null hypothesis. This means there is sufficient evidence to conclude that there is a significant difference in the mean time until the correct temperature is reached among the three refrigeration units.
* If the p-value is greater than or equal to 0.10, we fail to reject the null hypothesis. This means there is not enough evidence to conclude that there is a significant difference in the mean time.
2. **Boxplot:**
* The boxplot will visually represent the distribution of cooling times for each refrigeration unit.
* Significant differences in the medians and the spread of the boxes can suggest differences between the units.
**Conclusion:**
Based on the ANOVA results and the boxplot, you can determine whether there is a significant difference in the mean time until the correct temperature is reached by the three refrigeration units at the 10% level of significance.
**Note:**
* This analysis assumes that the data for each unit is normally distributed and that the variances of the three groups are equal. You may need to perform additional tests (e.g., normality tests, Levene's test for homogeneity of variances) to check these assumptions.
This analysis provides a framework for your investigation. You can then use the R output to draw conclusions about the effectiveness of the different refrigeration units.
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