Data On The Number Of Occurrences Per Time Period And Observ

Data On The Number Of Occurrences Per Time Period And Observed Freq

Data on the number of occurrences per time period and observed frequencies follow. Use α = .05 to perform the goodness of fit test to see whether the data fit a Poisson distribution. What hypotheses are appropriate for this test?

Data on the number of occurrences per time period and observed frequencies follow. Use α = .05 to perform the goodness of fit test to see whether the data fit a Poisson distribution. What test statistic is appropriate for this test?

Data on the number of occurrences per time period and observed frequencies follow. Use α = .05 to perform the goodness of fit test to see whether the data fit a Poisson distribution. What is the rejection region for this test?

Data on the number of occurrences per time period and observed frequencies follow. Use α = .05 to perform the goodness of fit test to see whether the data fit a Poisson distribution. Calculate the value of the test statistic. Use the "Number of Occurrences" as the categories.

Consider the goodness-of-fit test for a Poisson or Normal distribution. When the expected frequency in some category is less than 5, it is recommended that adjacent categories be combined to obtain expected frequencies that are all greater than 5. What is the reason for this recommendation?

The critical value for a 0.05 level of significance goodness-of-fit test for the Poisson distribution is the same as that for the Normal distribution. True or False?

Three different methods for assembling a product were proposed by an industrial engineer. To investigate the number of units assembled correctly with each method, 15 employees were randomly selected and assigned to the three methods, with 5 workers per method. The number of units assembled correctly was recorded, and ANOVA was applied. What hypotheses are appropriate for comparing the three groups?

Using the same scenario, what is the rejection region for the ANOVA test at an arbitrary significance level?

Based on the ANOVA table, what is the test statistic for the ANOVA test?

What is the ANOVA test statistic for a completely randomized design with three treatment groups?

Given the p-value of 0.004 for the ANOVA test, what conclusion is appropriate at the 0.05 significance level?

Using the means table and ANOVA table, with a t-value of 1.19, which pairs of groups show significant differences?

In a completely randomized design with three treatments, if the null hypothesis is rejected, what is the Bonferroni adjusted level of significance for multiple comparisons if the original significance level is 0.05?

In a randomized block design with three treatments and six blocks, the treatment sum of squares is 21 and block sum of squares is 30. What is the value of the ANOVA test statistic?

For a simple linear regression model with estimated line y = 60 + 5x and SSE = 1,530, with given data, how do you calculate the test statistic?

Paper For Above instruction

Introduction

Statistical tests are fundamental tools in analyzing data to determine underlying distributions, compare group means, and validate models. In contexts where data follow discrete distributions like the Poisson or continuous distributions such as the Normal, appropriate goodness-of-fit tests and comparisons help ascertain the suitability of these models. This paper discusses several statistical methods, focusing on the goodness-of-fit test for Poisson distributions, the analysis of variance (ANOVA) for comparing multiple groups, multiple comparison procedures, and linear regression hypothesis testing, providing detailed explanations, calculations, and interpretations.

Goodness-of-Fit Test for the Poisson Distribution

The primary hypotheses for assessing whether observed data conform to a Poisson distribution are: the null hypothesis (H₀), which states that the data follow a Poisson distribution; and the alternative hypothesis (H₁), which states that the data do not follow a Poisson distribution.

Mathematically, H₀: The data follow a Poisson distribution; H₁: The data do not follow a Poisson distribution.

The test statistic employed is the Chi-square goodness-of-fit test, calculated as:

χ² = Σ [(Oᵢ - Eᵢ)² / Eᵢ]

where Oᵢ is the observed frequency in the ith category, and Eᵢ is the expected frequency assuming a Poisson distribution. To calculate Eᵢ, we estimate the Poisson parameter λ using the sample mean and compute expected frequencies accordingly.

When performing the test, categories with expected frequencies less than 5 should be combined with adjacent categories to maintain test validity. This ensures the approximation to the Chi-square distribution remains accurate because the asymptotic properties of the test require sufficiently large expected counts.

The critical value for the Chi-square test at α = 0.05 depends on the degrees of freedom, typically calculated as the number of categories minus the number of estimated parameters minus one.

In the context of the same significance level, the critical value for the goodness-of-fit test for a Poisson distribution generally differs from that for a Normal distribution due to different degrees of freedom and distribution properties, making the statement that they are the same false.

Analysis of Variance (ANOVA)

ANOVA is used to compare the means across multiple groups. The hypotheses tested are: H₀ – All group means are equal; H₁ – At least one group mean differs.

The rejection region depends on the F-distribution, with degrees of freedom for numerator (between groups) and denominator (within groups). For a significance level α, the critical F-value can be obtained from F-tables or software.

The test statistic for ANOVA is calculated as:

F = MSB / MSW

where MSB is the mean square between groups, and MSW is the mean square within groups, derived from sums of squares divided by their respective degrees of freedom. Given the sums of squares, the test statistic is computed and compared to the critical F-value.

In the scenario where the p-value is less than 0.05 (such as 0.004), the null hypothesis is rejected, indicating significant differences among group means. Proceeding with multiple comparisons, such as pairwise t-tests with Bonferroni correction, helps identify which specific groups differ significantly.

The Bonferroni adjustment involves dividing the original significance level by the number of comparisons to control for Type I errors. For three groups, there are three pairwise comparisons, and the adjusted α is 0.05/3 ≈ 0.0167.

Using the t-value of 1.19, the significance of mean differences can be assessed by comparing the t-value to the critical t-value at the adjusted α, determining which pairs are significantly different.

Linear Regression Hypothesis Testing

In linear regression, hypotheses typically involve the slope coefficient, testing whether it differs significantly from zero. The test statistic is calculated as:

t = (b̃ - 0) / SE_b̃

where b̃ is the estimated slope, and SE_b̃ is its standard error. Given the regression line y = 60 + 5x and SSE = 1,530, the variance of the residuals can be used to estimate SE_b̃, which then allows the calculation of the t-statistic.

This hypothesis test determines whether the predictor variable significantly explains variability in the response variable, critical in assessing model validity.

Conclusion

Statistical hypothesis testing provides rigorous methods for analyzing data, from distribution fitting with goodness-of-fit tests to comparing group means with ANOVA, and assessing predictor significance in regression models. Proper application, including adherence to assumptions such as expected frequency thresholds and correct adjustment for multiple comparisons, ensures valid and reliable results in research settings.

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