Central Limit Theory

1.

Central Limit theory:

The central limit theorem states that when we have a large number of independent variables with an identical random distribution, the distribution of their sum tend to be normally distributed as the number of these variables increase.

Regarding other research:

The central theorem states that as the number of observation increases then the nature of the distributions seems to be normally distributed, therefore as we increase the number of observations then the distribution will tend to be normal, therefore it is important that we use a large number of observations in data analysis in order to achieve a normal distribution. Below is a normal distribution.

2.

Research question:

I choose the research question that depicts the economic relationship that exists between the demand of a product and the price, the demand of a good depends on the price of that good.

The independent variable and dependent variables:

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Central Limit theory

The dependent variable here is demand and the independent variable here will be price, therefore the model will be as follows:

Y = a + b X

Where Y is demand

a is the autonomous value

b is the slope

X is the price

One tail or two tail test:

The two tail test would be the best for the statistical test for this model. this is because when we use the two tail test will be in a position to check the statistical significance for the test.

Statistical hypothesis

The autonomous value

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Central Limit theory

Null hypothesis:

Ho: a = 0

Alternative hypothesis:

Ha: a ≠ 0

The slope

The autonomous value

Null hypothesis:

Ho: b = 0

Alternative hypothesis:

Ha: b ≠ 0

Alpha level

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Central Limit theory

Alpha level would be 0.05, this means that our level of test is 95%, this is a measure that can be used to construct an acceptable confidence interval, if we choose 0.01 then this means that our level of measure is 99% which due to the estimation of our model may lead to an acceptance of the null hypothesis. Therefore it would be best to use the 0.05 level of test.

3. We will consider crime rate and crime detection, in this case we will state the hypothesis that crime rate depends on the crime detection rate. If the crime detection rate is high then the crime rate will decrease.

The independent variable and the dependent variables

The independent variable here is crime detection. The dependent variable here will be crime rate.

Generate the statistical null and alternative hypotheses.

As in the case of the above example of demand and supply we will also consider a regression model that will have to have an autonomous value and a slope or the slope coefficient, there fore our model will be:

Y = a + b X

Where Y is crime rate

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Central Limit theory

a is the autonomous value

b is the slope

X is the crime detection

For the hypothesis statement

The autonomous value

Null hypothesis:

Ho: a = 0

Alternative hypothesis:

Ha: a ≠ 0

The slope

The autonomous value

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Central Limit theory

Null hypothesis:

Ho: b = 0

Alternative hypothesis:

Ha: b ≠ 0

The effect size and the probability value would not:

The effect side will tell you the nature of the relationship that exist between the two variable, for the probability value it will only give us the percentage of the dependent variable that is explained by the independent variable. Therefore the effect side will give us the relationship nature of the two variables, example whether the relationship is inverse or linear and also the nature of the signs.

Degrees of freedom sample size and t statistic

If the sample size is 10, then given that the numbers of variables are 2, then our degrees of freedom will be 10-2= 8, therefore when we test the null and alternative hypothesis our degrees of freedom will be 8. If we consider the T statistic value to be 0.05 at 8 degrees of freedom, then our critical value from the T table will be 1.859548. If our T calculated is greater than the t critical then we reject the null hypothesis.

4. The data to be used in this case is the test for a new drug regarding a new drug, when we consider a drug used to cure a certain drug we have to consider both samples where one does not use the drug and the other where the sample uses the drug, in this case we compare the two means of the drug significance and use the F distribution to determine the significance of

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Central Limit theory

the drug.

the ANOVA table analysis the various components of an estimated model, for a model there are RSS which is the random sum of squares, the ESS which the explained sum of squares and the TSS which is the total sum of squares, their relationship is asfolows:

Tss = Ess + Rss.

The anova table is as follows:

ss

d.f

mean ss

Ess

a

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Central Limit theory

2

a/2

Rss

b

n-3

a/n-3

Tss

c

n-1

a/n-1

The above analysis will use the F distribution whose purpose is to compare the difference in two means

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Central Limit theory

The independent variables and dependent variables

When we consider the drug use we will have to estimate the model on the significance of the drug, therefore the independent variable will be the use of the drug and the dependent variable will be the cure of the drug depending on the drug.

Hypotheses:

To compare the two means the hypothesis will be stated as follows:

Null hypothesis:

H0: b1=b2

Alternative hypothesis:

Ha: b1≠ b2

The effect size and the probability value

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Central Limit theory

The effect side will tell you the nature of the relationship that exist between the two variable, for the probability value it will only give us the percentage of the dependent variable that is explained by the independent variable. Therefore the effect side will give us the relationship nature of the two variables, example whether the relationship is inverse or linear and also the nature of the signs.

Additional information regarding confidence intervals

The confidence interval will provide us with provide us with the interval at which our estimated parameters will be varied at a given level of probability, at this interval this is the range at which the given parameter will be true.

Post hoc test:

A post hoc test will be used when the above hypothesis test is not statistically significant, it used as a follow up analysis when the F test and T test are not statistically significant, this test depends on the p value and the degrees of freedom as a way in which to verify the statistical significance.

References:

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