02 EJEMPLO ANOVA
Evaluating Temperature Effects on Production Rates
Introduction to the Experiment
- The discussion begins with a scenario in a production plant assessing whether temperature affects production rates, presenting data across three different temperatures.
- The objective is to determine if the temperature has a significant effect on production levels, using an alpha level of 0.05 for statistical significance.
Hypothesis Formation
- The null hypothesis states that the mean production rates at 20º, 22º, and 24º are equal, while the alternative hypothesis suggests that not all means are equal, indicating a significant temperature effect.
Assumptions Verification
- It is essential to verify assumptions such as random sampling, independence of samples, and normal distribution of factors before proceeding with analysis using an F-distribution.
Data Collection and Statistical Calculation
- Data from three samples corresponding to each temperature is collected; calculations will be based on this data.
- The total sum of squares (SS total), which quantifies variability in the data set, will be calculated using Excel.
Detailed Calculations
- The process involves squaring each observation and summing them up to derive necessary statistics for further analysis.
- SS total is partitioned into two components: SS factor (due to temperature levels) and SS error (due to experimental error), totaling 102.92.
Partitioning Sum of Squares
- Each component's calculation requires specific formulas; SS factor represents variability due to temperature changes while SS error accounts for replication errors.
- A detailed formulaic approach is used for calculating both sums of squares by considering sample sizes and squared values.
Finalizing Analysis Components
- After obtaining individual sums of squares (2.97 for temperature and 99.95 for error), these values confirm that they add up correctly to the total sum of squares calculated earlier.
Summary Table Creation
Understanding Degrees of Freedom in Statistical Analysis
Calculation of Degrees of Freedom
- The discussion begins with the calculation of degrees of freedom, where three columns are present, leading to a total of 2 degrees of freedom after subtracting one.
- Total degrees of freedom is calculated by taking the number of samples (13) and subtracting one, resulting in 12.
- The error degrees of freedom is determined by subtracting the number of columns from the sample size: 13 - 3 = 10.
Mean Square Calculations
- The mean square is derived from dividing the sum of squares for factors by their respective degrees of freedom. This applies similarly to errors.
- Using provided data, a mean square for factors is calculated as 42.25 and for error as 0.95, which are essential for further statistical analysis.
F-statistic Computation
- The F-statistic is computed by dividing the mean square factor (temperature) by the mean square error, yielding an F value of 44.47.
Probability Distribution and P-value Assessment
- Moving to probability distribution, a right-tail test is employed with an expression indicating that F > 44.47 with specified degrees of freedom.
- A table is referenced to estimate the P-value rather than obtaining a specific number; this helps in drawing conclusions about significance levels.
Conclusion on Hypothesis Testing
- It’s concluded that since P < 0.01, we reject the null hypothesis, indicating at least one temperature has a significant effect on production rates.
- Differences in average production rates across tested temperatures suggest they are statistically different from each other.