Apply the Kruskal-Wallis test and the following statistics to determine whether there is enough

Apply the Kruskal-Wallis test and the following statistics...

Key Concepts

Kruskal-Wallis Test

The Kruskal-Wallis test is a nonparametric method used to determine if there are statistically significant differences between the medians of three or more independent groups. It is an extension of the Mann-Whitney U test to multiple groups and does not assume normal distribution of the data, making it useful when the assumptions for ANOVA are not met.

Nonparametric Methods

Nonparametric methods, such as the Kruskal-Wallis test, are statistical techniques that do not rely on assumptions about the underlying population distribution. These methods are particularly effective when the sample sizes are small or when the data does not meet the requirements for classical parametric tests like homogeneity of variance or normality.

Hypothesis Testing

Hypothesis testing is a procedure for making inferences about populations based on sample data. It involves establishing a null hypothesis, which represents the status quo or no effect, and an alternative hypothesis, which indicates the presence of an effect or difference. The test then evaluates whether the observed data significantly contradict the null hypothesis.

Test Statistic Calculation Using Ranks

In the Kruskal-Wallis test, the test statistic is calculated based on the ranks of the data rather than the raw data values. All observations are ranked together, and these rank sums within each group are used to compute the statistic that is compared against a chi-squared distribution. This approach minimizes the influence of outliers and non-normality while assessing group differences.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used to determine whether an observed test statistic is sufficiently extreme to reject the null hypothesis. In the context of statistical tests such as the Kruskal-Wallis test, a 10% significance level indicates a 10% risk of rejecting the null hypothesis when it is actually true, thus controlling the probability of a Type I error.

Transcript

00:01 In this question, we are given with the data about two samples from two population.

00:07 The first sample is of size 6 to 1 and the second sample is of size 57.

00:12 The first, the standard deviation of first sample is 18 .3 and the standard deviation of the second sample is 13 .5.

00:20 We are asked to test whether the population standard deviation differs.

00:25 So we will take this statement as our alternative hypothesis, which is sigma 1 is not equal to sigma 2 it's different from sigma 2 and our null hypothesis will be sigma 1 is equal to sigma 2 alpha is given in the question which is 0 .05 we will use f statistics to test the to compare the standard deviation the variance of the two population so f statistics is the variance of first sample divided by variance of second sample with n1 minus 1 and n 2 minus 1 degree of freedom where n 1 minus 1 is degree of freedom of numerator and n2 minus 1 is degree of freedom of denominator so by putting the values in the f statistics we get the value is 13 1 .355 now we'll check our acceptance region the region where the way where when the value of f statistics falls we will accept our null hypothesis and the region beyond which the value will be rejected so it is two tail test so the lower a limit of the region will be f1 minus alpha divided by 2 with the degree of freedom is n1 minus 1 and n2 minus 2 and the upper limit will be f alpha divide by 2 and 1 minus 1.

01:47 So we'll check in the table.

01:50 We'll further solve it.

01:52 We will put the value alpha in it 0 .05 and divide it by 2 and then we subtract it from 1 for the lower limit...

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