Question

In: Math

Respond to all of the following questions in your posting for this week: Describe the characteristics...

Respond to all of the following questions in your posting for this week:

  • Describe the characteristics of the F distribution. Provide examples.
  • What are we testing when we test for two population variances? Explain your answer and provide an example.
  • What are the assumptions of an ANOVA, and when would you use an ANOVA?

Solutions

Expert Solution

Definition: The F-distribution depends on the degrees of freedom and is usually defined as the ratio of variances of two populations normally distributed and therefore it is also called as Variance Ratio Distribution

Properties of F-Distribution

There are several properties of F-distribution which are explained below:

The F-distribution is positively skewed and with the increase in the degrees of freedom ν1 and ν2, its skewness decreases.

The value of the F-distribution is always positive, or zero since the variances are the square of the deviations and hence cannot assume negative values. Its value lies between 0 and ∞.

The statistic used to calculate the value of mean and variance is:

The shape of the F-distribution depends on its parameters ν1 and ν2 degrees of freedom.

The values of the area lying on the left-hand side of the distribution can be found out by taking the reciprocal of F values corresponding to the right-hand side and the degrees of freedom in the numerator and the denominator are interchanged. This is called as Reciprocal Property of F-distribution. Symbolically, it can be represented as:This property is mainly used in the situations when the values of the lower tail F values are to be determined corresponding to the upper tail F values.

Thus, these are the properties of F-distribution that tells how the sample is distributed under study and what statistical inferences can be drawn therefrom.

Properties of F-Distribution

There are several properties of F-distribution which are explained below:

The F-distribution is positively skewed and with the increase in the degrees of freedom ν1 and ν2, its skewness decreases.

The value of the F-distribution is always positive, or zero since the variances are the square of the deviations and hence cannot assume negative values. Its value lies between 0 and ∞.

The statistic used to calculate the value of mean and variance is:Properties of F-distribution

The shape of the F-distribution depends on its parameters ν1 and ν2 degrees of freedom.

The values of the area lying on the left-hand side of the distribution can be found out by taking the reciprocal of F values corresponding to the right-hand side and the degrees of freedom in the numerator and the denominator are interchanged. This is called as Reciprocal Property of F-distribution. Symbolically, it can be represented as:roperties of F-distribution-2This property is mainly used in the situations when the values of the lower tail F values are to be determined corresponding to the upper tail F values.

Thus, these are the properties of F-distribution that tells how the sample is distributed under study and what statistical inferences can be drawn there fore

Example:

STATISTICS

Application Of F Distribution

   

In the estimation theory, we draw samples from the population and

The F Distribution is used by a researcher in order to carry out the test for the equality of the two population variances. If a researcher wants to test whether or not two independent samples have been drawn from a normal population with the same variability, then he generally employs the F-test. ANOVA is the best example depicting the use of F-test for comparing the variance ratio in which we find ratio F = variation between sample means/variation within the samples.An example depicting the above case in which the F-test is applied is, for example, if two sets of pumpkins are grown under two different experimental conditions. In this case, the researcher would select a random sample of size 9 and 11.The standard deviations of their weights are 0.6 and 0.8 respectively. After making an assumption that the distribution of their weights is normal, the researcher conducts an F-test to test the hypothesis on whether or not the true variances are equal.

Now the solution for this is:
We want to test H0: σ2x = σ2y

Against   H1: σ2x ≠ σ2y

Here n1= 11   n2 =9 sx=0.8   sy = 0.6

Under  H0,  F= Sx2/ Sy2 follows F(n1,n2)

Sx2 = (n1/n1+n2) sx2 =0.704

Sy2= (n1/n1+n2) sy2 =0.28125

Fcal= 0.704/0.28125 =   2.5  

Ftab (0.05) = 3.35 at (10,8) d.f

So Fcal < Ftab at 5% level of significance.

H0 may not be rejected, hence we can say that true variance is not equal. Or the samples of the pumpkins have come from the population having different variability with 95% confidence. F-test of equality of variances

is a test for the null hypothesis that two normal populations have the same variance. Notionally, any F-test can be regarded as a comparison of two variances,

F test is used test if the variances of two populations are equal. This test can be a two-tailed test or a one-tailed test. The two-tailed version tests against the alternative that the variances are not equal. The one-tailed version only tests in one direction, that is the variance from the first population is either greater than or less than (but not both) the second population variance

Let X1, ..., Xn and Y1, ..., Ym be independent and identically distributed samples from two populations which each have a normal distribution. The expected values for the two populations can be different, and the hypothesis to be tested is that the variances are equal. Let

be the sample means. Let

be the sample variances. Then the test statistic

has an F-distribution with n − 1 and m − 1 degrees of freedom if the null hypothesis of equality of variances is true. Otherwise it follows an F-distribution scaled by the ratio of true variances. The null hypothesis is rejected if F is either too large or too small based on the desired alpha level (i.e., statistical significance).

Example

A psychologist was interested in exploring whether or not male and female college students have different driving behaviors. The particular statistical question she framed was as follows:

Is the mean fastest speed driven by male college students different than the mean fastest speed driven by female college students?

The psychologist conducted a survey of a random n = 34 male college students and a random m = 29 female college students. Here is a descriptive summary of the results of her survey:

Is there sufficient evidence at the α = 0.05 level to conclude that the variance of the fastest speed driven by male college students differs from the variance of the fastest speed driven by female college students?

We're interested in testing the null hypothesis:

H0:σ2X=σ2YH0:σX2=σY2

against the alternative hypothesis:

HA:σ2X≠σ2YHA:σX2≠σY2

The value of the test statistic is:



F=12.21/20.1=0.368

Using the critical value approach, we divide the significance level α = 0.05 into 2, to get 0.025, and put one of the halves in the left tail, and the other half in the other tail. Doing so, we get that the lower cutoff value is 0.478 and the upper cutoff value is 2.0441

Because the test statistic falls in the rejection region, that is, because F = 0.368 ≤ 0.478, we reject the null hypothesis in favor of the alternative hypothesis. There is sufficient evidence at the α = 0.05 level to conclude that the population variances are not equal. Therefore, the assumption of equal variances that we made when performing the two-sample t-test on these data in the previous lesson does not appear to be valid

Assumptions for ANOVA

To use the ANOVA test we made the following assumptions:

Each group sample is drawn from a normally distributed population

All populations have a common variance

All samples are drawn independently of each other

Within each sample, the observations are sampled randomly and independently of each other

Factor effects are additive

The presence of outliers can also cause problems. In addition, we need to make sure that the F statistic is well behaved. In particular, the F statistic is relatively robust to violations of normality provided:

The populations are symmetrical and uni-modal.

The sample sizes for the groups are equal and greater than 10

In general, as long as the sample sizes are equal (called a balanced model) and sufficiently large, the normality assumption can be violated provided the samples are symmetrical or at least similar in shape

We now look at how to test for violations of these assumptions and how to deal with any violations when they occur.

Testing that the population is normally distributed (see Testing for Normality and Symmetry)

Testing for homogeneity of variances and dealing with violations (see Homogeneity of Variances)

Testing for and dealing with outliers (see Outliers in ANOVA)

When would we used an Anova

Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. It may seem odd that the technique is called "Analysis of Variance" rather than "Analysis of Means." ... ANOVA is used to test general rather than specific differences among means.

ANOVA is used to test general rather than specific differences among means. This can be seen best by example. In the case study "Smiles and Leniency," the effect of different types of smiles on the leniency shown to a person was investigated. Four different types of smiles (neutral, false, felt, miserable) were investigated. "All pairwise comparison among the means Showed how the test differences among the means.


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