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Use α = 0.01, and n = 100. Determine the Chi-Square value, and conclude accordingly regarding this goodness of fit test. All numbers are from a table of random numbers with each digit (0-9) having a probability of 1/10. Construct confidence intervals for σ² using given data, assuming the variable is normally distributed. Specifically:

- For part A: Use α = 0.01 to construct the confidence interval.

- For part B: Use α = 0.10 to construct the confidence interval.

- For part C: Explain why it was necessary to assume normal distribution for the variable.

Sample Paper For Above instruction

Introduction

The analysis of goodness of fit and the construction of confidence intervals are fundamental procedures in inferential statistics. These techniques allow researchers to test hypotheses about population distributions and estimate parameters such as variance with specified levels of confidence. This paper explores these concepts through the application of the Chi-Square goodness of fit test and confidence interval construction for the population variance (σ²), assuming the underlying variable follows a normal distribution.

Goodness of Fit Test using Chi-Square

The Chi-Square goodness of fit test assesses whether observed frequencies differ significantly from expected frequencies under a hypothesized distribution. Given n=100 observations drawn from a set of randomly generated digits (0-9), each with an equal probability of 1/10, the expected frequency for each digit is 10. The observed frequencies are compared to these expectations.

Using the observed counts (which would be provided from actual data), the Chi-Square statistic is calculated as:

χ² = Σ [(O_i - E_i)² / E_i]

where O_i represents observed frequency, and E_i represents expected frequency.

Since the data are from a random number table with digits uniformly distributed, the expected counts are consistent across categories, simplifying calculations. Under the null hypothesis that the digits are uniformly distributed, the Chi-Square statistic follows a Chi-Square distribution with degrees of freedom df = k - 1 = 9, where k=10.

Given α=0.01, the critical value from the Chi-Square distribution table at df=9 is approximately 21.666. If the calculated χ² exceeds this critical value, the null hypothesis of uniform distribution is rejected; otherwise, it is not.

Suppose, for example, the observed counts closely approximate expected counts, resulting in a χ² value below the critical threshold, implying no significant deviation from uniformity. This supports the hypothesis that the digits are uniformly distributed.

Constructing Confidence Intervals for σ²

Estimating the population variance (σ²) involves constructing confidence intervals based on sample data. Assuming the variable X is normally distributed, the sample variance s² can be used to generate these intervals. The formulas for the confidence interval are derived from the Chi-Square distribution:

For a confidence level 1 - α, the interval is:

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

where χ²_(α/2) and χ²_(1 - α/2) are the critical Chi-Square values at the respective tail probabilities, and n is the sample size.

Part A: α = 0.01

At a 99% confidence level, the critical values are obtained from the Chi-Square table at df=n-1. For n=100, df=99:

- χ²_(0.005) ≈ 124.342

- χ²_(0.995) ≈ 73.361

Thus, the confidence interval for σ² is:

[(99 × s²) / 124.342, (99 × s²) / 73.361]

This interval provides a narrow estimate of the population variance, reflecting high confidence due to the low alpha.

Part B: α = 0.10

At a 90% confidence level, the critical values are:

- χ²_(0.05) ≈ 113.145

- χ²_(0.95) ≈ 81.897

The confidence interval becomes:

[(99 × s²) / 113.145, (99 × s²) / 81.897]

The wider interval reflects increased uncertainty associated with a higher alpha.

Discussion: Why Normal Distribution Assumption is Essential

The assumption that the variable X is normally distributed is crucial because the derivation of the confidence intervals relies on properties of the Chi-Square distribution that are valid only under normality. Specifically, the distribution of the sample variance s² follows a scaled Chi-Square distribution exclusively when data are normally distributed. Violations of normality can lead to inaccurate confidence intervals, either underestimating or overestimating the true variance, and can affect the reliability of hypothesis tests like the Chi-Square goodness of fit.

In practice, many statistical procedures assume normality because of the Central Limit Theorem, which states that for sufficiently large sample sizes, the distribution of the sample mean (and, under certain conditions, variance) approximates normality. However, for constructing confidence intervals for variance, the normality assumption ensures the validity of the Chi-Square distribution application directly relating sample variance to the population variance.

Conclusion

The application of the Chi-Square goodness of fit test in assessing whether random number outputs conform to a uniform distribution demonstrated that, with adequate sample size and normality assumption, significant deviations can be detected effectively. Additionally, constructing confidence intervals for the population variance provides a quantifiable measure of uncertainty, with the confidence level directly impacting the interval's width. The assumption of normality remains fundamental in these procedures, underpinning the theoretical foundation for inference about variance. Proper understanding and application of these techniques enable robust statistical analysis and inference in various research contexts.

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