**Exercise 22 Explanation for Educational Website:**This exercise involves modifying data from a previous problem (Problem 21) by doubling the sample size to $ n = 10 $ while keeping the same sample means and variances. The task is to analyze how this change impacts the F-ratio using ANOVA. The data is presented in a table divided into three treatments (I, II, and III).**Table Layout:**- Each treatment has the following parameters: - **n**: Sample size $ (n = 10) $ - **M**: Sample mean - **T**: Total sum of scores in the treatment - **s²**: Sample variance - **SS**: Sum of squares for the treatment**Treatment Details:**- **Treatment I:** - $ n = 10 $ - $ M = 1 $ - $ T = 10 $ - $ s² = 9.00 $ - $ SS = 81 $- **Treatment II:** - $ n = 10 $ - $ M = 5 $ - $ T = 50 $ - $ s² = 10.00 $ - $ SS = 90 $- **Treatment III:** - $ n = 10 $ - $ M = 6 $ - $ T = 60 $ - $ s² = 11.00 $ - $ SS = 99 $**Overall Data:**- Total sum of all treatments ($ \Sigma X² = 890 $)- $ N = 30 $ (Total number of data points across all treatments)- Grand total ($ G = 120 $)**Exercise Requirement:**a. **Prediction and Analysis:** - Predict how increasing the sample size affects the F-ratio for these data compared to the values from Problem 21. - Use an ANOVA with $ \alpha = 0.05 $ to test this prediction. - Note: Since the samples are of the same size, $ MS_{within} $ (Mean Square Within) is the average of the three sample variances.This exercise provides a hands-on understanding of how sample size influences statistical analysis using the ANOVA method.

I II III n=10 n=10 n=10 M=1 M=5 M=6 N=30 T=10 T=50 T=60 G=120 s2=9.00 s2=10.00 s2=11.00 ∑Xi2=890 SS=81 SS=90 SS=99 level of significance α=0.05 The increase in the sample size increases the value of the F-ratio. Since a large sample size the smaller differences in means. Now to state the null hypothesis and alternate hypothesis is given below, H0:μ1=μ2=μ3All treatments have equal effectsH1:Atleast one of the treatment mean is different Now, dfbetw=k-1=3-1dfbetw=2now to find dfwithin,dfwithin=9+9+9dfwithin=27 Here, G=10+50+60 G=120 Now to find SSbetw,SSbetw=∑nixi2-G2N=10×12+52+62-120230=10×1+25+36-1440030=10×62-480=620-480=140SSbetw=140 To calculate MSbetw,MSbetw=SSbetwdfbetw=1402=70to calculate MSwithin,MSwithin=9.00+10.00+11.003=303=10 To calculate F statistic as below,F=MSbetwMSwithin=7010=7Fcritic=Fdf1,df2,α2=F0.025,2,27=4.242F>Fcritic i.e. 7>4.242 i.e. H0 is rejected at a 0.05 level of significance. We reject H0 means we accept Ha i.e. atleast one of the treatmenrt means is different.

Answered: 22. To create the following data we…

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Statistics

22. To create the following data we started with the same sample means and variances that appeared in problem 21 but doubled the sample size to n 10. Treatment

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

**Exercise 22 Explanation for Educational Website:**

This exercise involves modifying data from a previous problem (Problem 21) by doubling the sample size to $ n = 10 $ while keeping the same sample means and variances. The task is to analyze how this change impacts the F-ratio using ANOVA. The data is presented in a table divided into three treatments (I, II, and III).

**Table Layout:**

- Each treatment has the following parameters:
- **n**: Sample size $ (n = 10) $
- **M**: Sample mean
- **T**: Total sum of scores in the treatment
- **s²**: Sample variance
- **SS**: Sum of squares for the treatment

**Treatment Details:**

- **Treatment I:**
- $ n = 10 $
- $ M = 1 $
- $ T = 10 $
- $ s² = 9.00 $
- $ SS = 81 $

- **Treatment II:**
- $ n = 10 $
- $ M = 5 $
- $ T = 50 $
- $ s² = 10.00 $
- $ SS = 90 $

- **Treatment III:**
- $ n = 10 $
- $ M = 6 $
- $ T = 60 $
- $ s² = 11.00 $
- $ SS = 99 $

**Overall Data:**

- Total sum of all treatments ($ \Sigma X² = 890 $)
- $ N = 30 $ (Total number of data points across all treatments)
- Grand total ($ G = 120 $)

**Exercise Requirement:**

a. **Prediction and Analysis:**
- Predict how increasing the sample size affects the F-ratio for these data compared to the values from Problem 21.
- Use an ANOVA with $ \alpha = 0.05 $ to test this prediction.
- Note: Since the samples are of the same size, $ MS_{within} $ (Mean Square Within) is the average of the three sample variances.

This exercise provides a hands-on understanding of how sample size influences statistical analysis using the ANOVA method.

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