Stat 3300 Homework 8 Due Wednesday 05272020 ✓ Solved

Stat 3300 Homework 8 due Wednesday 05272020

Stat 3300 Homework 8 due Wednesday 05272020

Identify the key assignment: solving statistical problems related to ANOVA, degrees of freedom, p-values, and experimental design. The tasks include calculating degrees of freedom, identifying variables, interpreting ANOVA tables, and making conclusions based on F-tests and p-values.

Sample Paper For Above instruction

Introduction

Understanding the principles of ANOVA (Analysis of Variance) is essential for comparing multiple groups and making inferences about their differences. This paper explores five rigorous statistical problems involving degrees of freedom, p-values, and experimental design considerations in the context of real-world research scenarios.

Question 1: Degrees of Freedom and P-Values

The first task involves calculating degrees of freedom (df) for various group comparisons, followed by the computation of p-values using R. The scenarios include comparing six groups with five observations each, four groups with eleven observations, and five groups across a total of 65 observations.

For example, in (a), with six groups each having five observations, the total number of observations is 30. The degrees of freedom for numerator (between-group) is c - 1 = 6 - 1 = 5, and for denominator (within-group) is N - c = 30 - 6 = 24. Utilizing R’s pf() function, the p-value for F=2.47 with df1=5 and df2=24 can be computed as p = 1 - pf(2.47, 5, 24), which yields approximately 0.072.

Similarly, in (b), with four groups and 11 observations each, total observations are 44, dfs are 3 and 40, and the p-value for F=5.03 is calculated accordingly. In (c), with total observations 65 across five groups, dfs are 4 and 60, and the p-value for F=3.11 is obtained.

Question 2: Experimental Design Variables and Group Structures

This problem addresses the identification of response variables, population comparisons, and group counts for three experimental contexts:

a) The effectiveness of VR navigation methods: response is the time in seconds to complete a path. The population comprises children assigned randomly to four groups: joystick, wand, dance mat, and gesture-based gloves; each group has 10 children, totaling N=40, with c=4 groups and ni=10 per group.

b) The effect of pesticides on bird calcium: the response is calcium content in mg. The population includes chicks allocated to 5 diet groups, each with 13 individuals, totaling N=65, c=5, and ni=13.

c) The impact of free food promotions on sales: the response is sales during a time window. The 20 weekdays are divided into groups where customers are offered different freebies or no freebie, each group appearing five times, leading to c=4, each with ni=5, and N=20 days.

Question 3: ANOVA Analysis on Psychiatrists' Perceptions

Given an ANOVA table, estimate the total number of psychiatrists, compute the common standard deviation, determine the p-value, and draw conclusions:

a) Total number of psychiatrists = df_total + 1 (since degrees of freedom sum up to total N-1). Here, df_total is 0.32, which appears inconsistent; probably a typographical error is present. Assuming the table outputs are correctly interpreted, the total N equals the df of age + df of error + 1, leading to N= number of psychiatrists.

b) The estimated common standard deviation (σ) is √MS_error, where MS_error = 153.32, thus σ≈ √153.32 ≈ 12.39.

c) The p-value is based on F = 0.45 with df1=2 and df2=0.54 (which is nonsensical, indicating a possible typo). Assuming correction, with df1=2 and df2= error df, p-value is obtained via R's pf().

d) Since the F statistic is low (0.45), and the p-value is high (above 0.05), the conclusion is to fail to reject the null hypothesis, indicating no significant difference in perceptions among age groups.

Question 4: ANOVA on Haptic Feedback in Gaming

Using group means, standard deviations, and sample sizes, the ANOVA table is constructed:

| Group | Mean | SD | n |

|--------------------------------|--------|-------|------|

| Standard joystick | X1 | S1 | 20 |

| Force feedback joystick | X2 | S2 | 20 |

| Vibration feedback joystick | X3 | S3 | 20 |

Calculations involve computing the overall mean, sum of squares between, error, and total, from which the F statistic is derived.

The degrees of freedom between groups = c - 1 = 2, and within groups = N - c = 60. The F value is then compared to the critical F value. The p-value obtained from R indicates whether there are statistically significant differences among joystick types, leading to conclusions about the effectiveness of haptic feedback.

Conclusion

The analysis demonstrates how degrees of freedom influence p-value calculations, which ultimately inform decision-making regarding group differences. Proper experimental design and statistical interpretation are crucial for drawing valid conclusions from research data.

References

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