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Please help me to answer these questions thanks. 1-What are the characteristics of standard normal distribution? The HR department of an organization collects data on employees: age, salary, level of education, gender, and ethnicity. Which data do you think is more likely to follow normal distribution? Explain why 2-The Rule of Three generates a 95% confidence interval. What rule would you recommend if you wanted to have a 99.75% confidence interval? Why?

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The characteristics of the standard normal distribution are foundational concepts in statistics that describe a specific type of probability distribution. This distribution is symmetric, bell-shaped, and centered around its mean, with a mean of zero and a standard deviation of one. Its key characteristics include symmetry about the mean, the total area under the curve equaling one, and the empirical rule, which states that approximately 68% of data falls within one standard deviation, 95% within two, and 99.7% within three. These properties make the standard normal distribution a crucial reference point for many statistical analyses.

The standard normal distribution is particularly useful because many natural and social phenomena tend to produce data that approximate this distribution when sample sizes are large enough. It allows statisticians to perform hypothesis testing, construct confidence intervals, and calculate probabilities efficiently. The symmetry and known properties enable the use of z-scores, which standardize individual data points relative to the distribution's mean and standard deviation, simplifying comparisons across different datasets.

Regarding the HR department’s data collection, certain variables are more likely to follow a normal distribution than others. Age, for example, often approximates normality in a broad population due to biological and societal factors influencing age distribution. Most individuals tend to cluster around a median age with fewer individuals significantly younger or older, resulting in a bell-shaped distribution. Salary, however, may not follow a normal distribution, as it often exhibits a right-skewed pattern with a long tail toward higher incomes, driven by a small number of employees earning significantly more than the average. Level of education might sometimes follow a normal distribution if data are aggregated across large populations, but it can also be categorical or ordinal, making it less ideal for modeling with a normal distribution. Gender and ethnicity are categorical variables and do not follow any continuous probability distribution like the normal; instead, they are better analyzed with categorical data techniques.

The Rule of Three is a widely-used heuristic in statistics that provides an approximate 95% confidence interval for the probability in cases of zero observed events. It states that if no events are observed in n trials, then the probability of the event occurring is at most 3/n with 95% confidence. This rule is particularly useful in quality control and rare event estimation where no failures occur in a sample size.

To achieve a higher confidence level, such as 99.75%, adjustments must be made to the rule used to encompass the increased certainty. A logical extension is to employ the inverse of the binomial probability or adjust the multiplier to reflect the new confidence level. Specifically, confidence intervals based on a Poisson approximation or the Wilson score interval can be used, with corresponding critical values from the normal distribution. For a 99.75% confidence level, the z-score is approximately 3.00, as opposed to 1.96 for 95%. The generalized rule would suggest that if no events are observed in n trials, then the probability p of the event is at most approximately 3×z²/n, where z is the z-score for the desired confidence level. Therefore, for 99.75%, the rule adjusts to a multiplier of approximately 3×(3.00)²/n, which is roughly 27/n, providing a conservative upper bound. This ensures a higher degree of confidence in the estimate, aligning with the increased confidence interval.

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