In A Completely Randomized Design 10 Experimental Units

In A Completely Randomized Design 10 Experimental Units

In a completely randomized design, 10 experimental units were used for the first treatment, 12 for the second treatment, and 19 for the third treatment. Sum of Squares due to Treatments and Sum of Squares Total is computed as 1100 and 1700 respectively. Prepare the ANOVA table and complete the same (fill out all the cells). State the Hypotheses. At a 0.05 level of significance, is there a significant difference between the treatments? Use both p-Value and Critical-Value approaches.

A random sample of monthly gasoline bills for a company's 15 sales persons are: what is the mean? what is the median? what is the value of quartile 3? what is the value of the mode? the range is equal to? what is the value of the standard deviation? what is the value of the coefficient of variation? do not express the answer as a percent, leave it as a ratio, e.g., 0.123. Are the numbers skewed positively, negatively, or are they symmetrical? (For your answer enter only one word: positively, negatively, or symmetrical.) What is the value of the coefficient of skewness using Pearson's coefficient of skewness? What is the value of quartile 2?

Paper For Above instruction

The assignment encompasses two primary statistical analyses: an ANOVA (Analysis of Variance) for a completely randomized design with three treatments, and descriptive statistics for a sample of gasoline bills for 15 salespersons. This comprehensive approach aims to assess the significance of treatment differences and to characterize the central tendency, dispersion, and distribution shape of the gasoline bills.

Analysis 1: ANOVA for a Completely Randomized Design

The provided data indicates that three treatments were applied across different experimental units: 10 units for the first treatment, 12 for the second, and 19 for the third. The sum of squares for treatments (SS_t) is 1100, and the total sum of squares (SS_total) is 1700. To analyze whether these treatments significantly differ in their effects, an ANOVA table is constructed.

The hypotheses for this analysis are formulated as follows:

• Null hypothesis (H₀): All treatment means are equal (no treatment effect).
• Alternative hypothesis (H₁): At least one treatment mean differs.

Calculating degrees of freedom (df):

• df for treatments (df_t): k - 1 = 3 - 1 = 2
• df for total (df_total): N - 1 = (10 + 12 + 19) - 1 = 41 - 1 = 40
• df for error (df_error): df_total - df_t = 40 - 2 = 38

The mean square for treatments (MS_t) is calculated as SS_t / df_t = 1100 / 2 = 550.

The mean square for error (MS_error) is calculated as (SS_total - SS_t) / df_error = (1700 - 1100) / 38 = 600 / 38 ≈ 15.79.

Constructing the ANOVA table:

To determine if the treatments differ significantly, compare the F-statistic (≈34.81) with the critical F-value from F-distribution tables at α = 0.05 with (2, 38) degrees of freedom. The critical F-value is approximately 3.24. Since 34.81 > 3.24, we reject H₀, indicating significant treatment effects.

Alternatively, using the p-value approach, the p-value associated with F(2, 38) = 34.81 is less than 0.001, confirming the significance at the 0.05 level.

Analysis 2: Descriptive Statistics of Gasoline Bills

For the gasoline bills of 15 salespersons, the data must be summarized through various statistics:

1. Mean

The mean (μ) is calculated as the sum of all observations divided by 15. Assuming the total sum of bills is S, then μ = S/15. For example, if the sum of the bills is 1500 units, then the mean is 1500/15 = 100.

2. Median

The median is the middle value when the 15 bills are ordered from smallest to largest. Since 15 is odd, the median is the 8th value in the ordered dataset.

3. Quartile 3 (Q3)

Q3 is the 75th percentile, found at the position 0.75(n+1) = 0.7516 = 12th value in the ordered data.

4. Mode

The value that appears most frequently in the dataset. If multiple modes exist, note all.

5. Range

The range is the difference between the maximum and minimum values in the dataset.

6. Standard Deviation

The standard deviation measures dispersion around the mean, calculated as the square root of the variance, which is the average squared deviation from the mean.

7. Coefficient of Variation (CV)

The CV is the ratio of the standard deviation to the mean, providing a standardized measure of dispersion without expressing as a percentage: CV = standard deviation / mean.

8. Distribution Skewness

Skewness indicates the asymmetry of the distribution. If skewness is close to zero, distribution is symmetrical; positive skew indicates a longer tail to the right; negative skew indicates a longer tail to the left.

9. Coefficient of Skewness using Pearson's coefficient

Pearson's skewness coefficient can be computed using the formula: (mean - mode) / standard deviation, or alternatively, using the sample skewness formula.

10. Quartile 2 (Q2)

Quartile 2 is the median, as the second quartile divides the dataset into two equal halves.

In conclusion, performing these calculations allows a thorough understanding of the data distribution and variability, which can be compared to assess whether the gasoline bills are skewed positively, negatively, or are symmetrical.

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