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# 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|STAT577 Evaluating the Normality of the Univariate Marginal Distributions

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## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Evaluating the Normality of the Univariate Marginal Distributions

Dot diagrams for smaller $n$ and histograms for $n>25$ or so help reveal situations where one tail of a univariate distribution is much longer than the other. If the histogram for a variable $X_{i}$ appears reasonably symmetric, we can check further by counting the number of observations in certain intervals. A univariate normal distribution assigns probability $.683$ to the interval $\left(\mu_{i}-\sqrt{\sigma_{i i}}, \mu_{i}+\sqrt{\sigma_{i i}}\right)$ and probability $.954$ to the interval $\left(\mu_{i}-2 \sqrt{\sigma_{i i}}, \mu_{i}+2 \sqrt{\sigma_{i i}}\right)$. Consequently, with a large sample size $n$, we expect the observed proportion $\hat{p}{i 1}$ of the observations lying in the interval $\left(\bar{x}{i}-\sqrt{s_{i i}}, \bar{x}{i}+\sqrt{s{i i}}\right)$ to be about .683. Similarly, the observed proportion $\hat{p}{i 2}$ of the observations in $\left(\bar{x}{i}-2 \sqrt{s_{i i}}, \bar{x}{i}+2 \sqrt{s{i i}}\right)$ should be about 954 . Using the normal approximation to the sampling distribution of $\hat{p}{i}$ (see [9]), we observe that either $$\left|\hat{p}{i 1}-.683\right|>3 \sqrt{\frac{(.683)(.317)}{n}}=\frac{1.396}{\sqrt{n}}$$
or
$$\left|\hat{p}_{i 2}-.954\right|>3 \sqrt{\frac{(.954)(.046)}{n}}=\frac{.628}{\sqrt{n}}$$
would indicate departures from an assumed normal distribution for the $i$ th characteristic. When the observed proportions are too small, parent distributions with thicker tails than the normal are suggested.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|A Q-Q plot for radiation data

The quality-control department of a manufacturer of microwave ovens is required by the federal government to monitor the amount of radiation emitted when the doors of the ovens are closed. Observations of the radiation emitted through closed doors of $n=42$ randomly selected ovens were made. The data are listed in Table $4.1$ on page 192 .

In order to determine the probability of exceeding a prespecified tolerance level, a probability distribution for the radiation emitted was needed. Can we regard the observations here as being normally distributed?

A computer was used to assemble the pairs $\left(q_{(j)}, x_{(j)}\right)$ and construct the $Q-Q$ plot, pictured in Figure $4.6$ on page 192. It appears from the plot that the data as a whole are not normally distributed. The points indicated by the circled locations in the figure are outliers-values that are too large relative to the rest of the observations.

For the radiation data, several observations are equal. When this occurs, those observations with like values are associated with the same normal quantile. This quantile is calculated using the average of the quantiles the tied observations would have if they all differed slightly.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代 考|Evaluating the Normality of the Univariate Marginal Distributions

[9])，我们观察到
$$|\hat{p} i 1-.683|>3 \sqrt{\frac{(.683)(.317)}{n}}=\frac{1.396}{\sqrt{n}}$$

$$\left|\hat{p}{i 2}-.954\right|>3 \sqrt{\frac{(.954)(.046)}{n}}=\frac{.628}{\sqrt{n}}$$ 将表明偏离假设的正态分布 $i$ 特性。当观察到的比例太小时，建议使用尾部比正态更粗的父分布。

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