Two DQ and one activity questions. Dissertation correction.

Numerical Example

Presume one randomly selects 7 girls from a population of girls and 12 boys from a population of boys. Below is a standard deviation in each sample population.

Calculate the f statistic.

Solution:

σ1 -Population 1 standard deviation

s1 - standard deviation of the drawn sample from population 1

σ2 -Population 2 standard deviation

s2 - standard deviation of the drawn sample from population 2

f = [ s12/σ12 ] / [ s22/σ22 ]

If the girls' data is evident in the numerator, the f statistic is calculated as follows:

f = ( 3502 / 3002 ) / ( 4502 / 5002 )

f = (122,500 / 90,000) / (202,500 / 250,000)

f = 1.361 / 0.81 = 1.68

The Degree of Freedom:

Numerator - v1 = 7 - 1 = 6

Denominator - v2 = 12 - 1 = 11

Additionally, if the men’s data is on the numerator:

f = ( 4502 / 5002 ) / ( 3502 / 3002 )

f = (202,500 / 250,000) / (122,500 / 90,000)

f = 0.81 / 1.361 = 0.595

The Degree of Freedom:

Numerator - v1 = 12 – 1 = 11

Denominator - v2 = 7 – 1 = 6

Between and Within-group Difference in an Anova

In ANOVA, within-group difference is also known as error variance or error group and refers to variations due to differences within individual levels or groups. That is, values within each group differ (Schmeckenbecher, 2020). In comparison, between-groups are trials with two or more subject groups tested concurrently by an alternate testing factor.

Numerical Example

A research company has four groups representing drugs A B C D, each composed of 200 individuals in each group. They measure their cholesterol levels. Calculate the between and within-group variation.

Calculation

The Sum of Squares within groups links intrinsically between-group variations, which is the Sum of Squares between variance differences due to the interaction of the groups with each other. ANOVA aims to compare the ratio within and between-group variance. If variations from interactions between different samples are much more significant than variations inside values in a single group, the means are unequal.

Difference between Covariance and Correlation

Their differences are as follows. Firstly, covariance is a measure that indicates the extent to which two random variables change in tandem. In comparison, correlation is a measure that represents the strong relationship between two random variables (Kumar, 2017). Secondly, covariance measures the correlation between -1 and +1, while correlation is a scaled covariance form lying between -∞ and +∞ (Kumar, 2017). Thirdly, covariance changes in scale; if the values of one variable are multiplied by a constant and the values of an alternate valuable are multiplied by a different or similar constant, the covariance alters. In comparison, correlation is unaltered by alterations in scale (Kumar, 2017). Finally, the correlation has no dimensions, while the covariance has dimensions as we obtain a value by the product of the units of two variables (Kumar, 2017).

Numerical Example

Calculate the covariance for the following:

Calculation:

Mean:

x = (2 + 4 + 6 + 8) / 4 = 5

y = (5 + 10 + 15 + 20) / 4 = 12.5

Cov(x, y) = ΣE ((x-μ) (y-ν)) / n-1

(2 - 5) (5 - 12.5) + (4 - 5) (10 – 12.5) + (6 - 5) (15 - 12.5) + (8 - 5) (20 - 12.5) = (-3) (-7.5) + (-1) (-2.5) + (1) (2.5) + (3) (7.5) =

(22.5 + 2.5 +2.5 + 22.5) / (4-1)

50 / 3 = 16.7

Correlation Numerical Example

Find the correlation coefficient from the data below:

Calculation:

r = n (Σxy) – (Σx) (Σy) / [√ [[n (ΣX2) – (Σx) 2] × [n (Σy2) – (Σy) 2]]

Σx = 20

Σy = 155

Σxy = 700

Σx2 = 120

Σy2 = 11,725

n is the sample size = 4

The correlation coefficient =

4(700) – (20) (155) / [√ [[4(120) – (20) 2] × [4(11,725) – (155) 2]]] =

(2800 – (3100) / [√ (480 – 400) × (46,900 – 24,025)]

(-300) / (√ (80 × 22,875) = -300 / 1352.78 = -0.2218

The range of the correlation coefficient is from -1 to 1. Our result is -0.2218 or 22.18%, which means the variables have a moderate positive correlation.

References

Kumar, P. (2017, August 24). Covariance vs. Correlation. LinkedIn. Derived from https://www.linkedin.com/pulse/covariance-vs-correlation-kumar-p

Schmeckenbecher, Y. (2020, May 9). What is the difference between a between groups and a within groups Anova? Asking Lot. Derived from https://askinglot.com/what-is-the-difference-between-a-between-groups-and-a-within-groups-anova

Surbhi, S. (2018, February 10). Difference between T-test and F-test. Key Differences. Derived from https://keydifferences.com/difference-between-t-test-and-f-test.html