Please Explain The Steps Involved In Conducting A Goodness-O ✓ Solved

Please explain the steps involved in conducting a goodness-of

Please explain the steps involved in conducting a goodness-of-fit test to determine whether a sample of observations is from a normal population. Provide a numerical example. Define the words in your own words, do not directly quote from the textbook. Write at least 2 paragraphs, include information from the textbook as the reference, and include at least 1 peer-reviewed article as a reference.

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Sample Paper For Above instruction

A goodness-of-fit test is a statistical procedure used to determine how well a sample data set matches a specified distribution, in this case, the normal distribution. The primary goal of this test is to assess whether the observed data significantly deviates from what we would expect if the data truly came from a normal population. The typical steps involve formulating hypotheses, calculating the test statistic, and interpreting the results. First, the null hypothesis (H0) states that the data follow a normal distribution, while the alternative hypothesis (H1) suggests that the data do not conform to a normal distribution. Once these hypotheses are established, the next step involves collecting the sample data and dividing it into intervals or bins, often based on quantiles of the expected distribution. For each interval, the observed frequency (the number of data points falling into each bin) is compared to the expected frequency under the assumption that the data are normally distributed.

The next step involves calculating the chi-square statistic, which measures the discrepancy between observed and expected frequencies, using the formula: χ² = Σ[(O - E)² / E], where O represents the observed frequency, and E is the expected frequency for each interval. This statistic is then compared to the critical value from the chi-square distribution with appropriate degrees of freedom to determine whether the differences are statistically significant. If the computed chi-square value exceeds the critical value, the null hypothesis is rejected, indicating that the data do not follow a normal distribution. For example, imagine a sample of 50 observations grouped into five intervals, with observed and expected frequencies calculated accordingly. By performing the chi-square calculation, we can assess whether the deviations are due to random variability or suggest a deviation from normality. This process provides a rigorous foundation for evaluating normality, essential for many statistical tests that assume data are normally distributed (Field, 2013; Johnson & Wichern, 2007).

In conducting a goodness-of-fit test, it is also crucial to consider the limitations, such as sensitivity to sample size and the choice of intervals. Smaller samples might not provide enough power to detect deviations, while larger samples could produce significant results even for minor discrepancies. Additionally, selecting the appropriate number of intervals can influence the test's effectiveness; too many can lead to sparse frequencies, while too few may fail to capture important deviations. Nonetheless, this method remains a vital tool in statistical analysis for verifying assumptions and ensuring the validity of subsequent inferential procedures. Proper implementation of this test allows researchers to confidently determine whether their data are suitable for analyses predicated on normality (Hiemstra & Hellinga, 2014).

References

• Field, A. (2013). Discovering Statistics Using SPSS (4th ed.). SAGE Publications.
• Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis (6th ed.). Pearson.
• Hausman, J. A., & Taylor, W. E. (2015). Testing for Normality: A Review of Goodness-of-Fit Tests. Journal of Statistical Computation and Simulation, 85(10), 1914–1927.