Use A01 And N100 To Determine The Chi-Square Value 085271

Use Α01 And N100determine The Chi Square Value And Come To The

Determine the Chi-Square value using α = 0.01, N = 100, and the provided observed and expected values. Based on the calculated Chi-Square statistic and the corresponding critical value from the chi-square distribution table at the 0.01 significance level, conclude whether there is evidence to reject the null hypothesis of goodness of fit. Additionally, construct confidence intervals for σ² using given variables, assuming normal distribution, at α = 0.01 and α = 0.10, and explain why the assumption of normality is necessary.

Paper For Above instruction

The Chi-Square test for goodness of fit is a fundamental statistical procedure used to determine whether observed categorical data fit an expected distribution. In the context of this problem, with a sample size of N=100, the goal is to calculate the Chi-Square test statistic and compare it to the critical value at a significance level of α=0.01. This comparison helps in making an informed decision about whether the observed data significantly deviate from the expected distribution, thus indicating if the null hypothesis should be rejected.

To perform this calculation, one must first determine the observed frequencies and expected frequencies for each category. For instance, if the categories are numbered from 0 to 9, and the expected probability for each is 1/10, then the expected count per category is N/10, or 10. The observed counts are derived from the data, while the expected counts are based on the hypothesized distribution. The Chi-Square statistic is then computed as the sum over all categories of the squared difference between observed and expected counts, divided by the expected counts:

χ² = Σ [(O_i - E_i)² / E_i]

Once the Chi-Square value is obtained, it is compared to the critical value from the chi-square distribution table with degrees of freedom equal to (number of categories - 1). For example, with 10 categories, df=9. If the calculated χ² exceeds the critical value at α=0.01 (which is approximately 21.666 for df=9), the null hypothesis is rejected, indicating the observed data do not fit the expected distribution well.

Beyond hypothesis testing, constructing confidence intervals for σ² provides insight into the variability of the data. Assuming the variable is normally distributed, the confidence interval for the population variance σ² is based on the chi-square distribution of the sample variance. Given a sample variance s², the 100(1 - α)% confidence interval for σ² is:

[( (n-1)s² ) / χ²_(1-α/2, n-1) , ( (n-1)s² ) / χ²_(α/2, n-1) ]

At α=0.01, the chi-square critical values are obtained from the chi-square table for df=n-1. Similarly, for α=0.10, these values are different and typically larger or smaller depending on the tail. These intervals provide a range within which the true population variance lies with specified confidence levels. The assumption of normality is critical because the derivation of the confidence interval for σ² relies on the distribution of the sample variance, which follows a chi-square distribution only when the underlying data are normally distributed.

The normality assumption ensures the validity of the chi-square distribution of the sample variance. Without this assumption, the distributional properties used in constructing accurate confidence intervals are compromised, which could lead to inaccurate inferences about the population variance. Consequently, verifying normality is essential before applying these intervals, often done through graphical methods or normality tests.

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