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Let the ordered sample observations be denoted by \({y_1}, {y_2}, \ldots , {y_n}\)(\({y_1}\) being the smallest and \({y_n}\)the largest). Our suggested check for normality is to plot the \({\phi ^{ - 1}}((i - .5)/n),{y_i})\)pairs. Suppose we believe that the observations come from a distribution with mean, and let \({w_1}, \ldots , {w_n}\)be the ordered absolute values of the \({x_i}'s\). A half 颅normal plot is a probability plot of the\({w_i}'s\). More specifically, since \(P\left( {|Z| \le w} \right) = P\left( { - w \le Z \le w} \right) = 2\phi \left( w \right) - 1\), a half-normal plot is a plot of the \(\left( {{\phi ^{ - 1}}/ \left\{ {\left( {\left( {i - .5} \right)/n + 1} \right)/2} \right\},{w_i}} \right)\)pairs. The virtue of this plot is that small or large outliers in the original sample will now appear only at the upper end of the plot rather than at both ends. Construct a half-normal plot for the following sample of measurement errors, and comment: \(23.78, 21.27,\)\(1.44, 2.39, 12.38, \)\(243.40, 1.15, 23.96, 22.34, 30.84.\)

Step by step solution

01

Definition of Probability

The probability of an event occurring is defined by probability. There are many scenarios in which we must forecast the outcome of an event in real life. The outcome of an event may be certain or uncertain. In such instances, we say that the event has a chance of happening or not happening.

02

Explanation for the half-normal plot.

First, we calculate the\({p_i}\)-values corresponding to the given sample values.

\({p_i} = \frac{{\left( {\frac{{i - 0.5}}{n}} \right) + 1}}{2}\)

Here, 鈥榥鈥� denotes the total number of sample values. In our case:\(n = 10\)

From the given\({p_i}\)-values, we can calculate corresponding z-scores as:

\({z_i} = {\phi ^{ - 1}}\left( {{p_i}} \right)\)

The table given below lists all the ordered absolute values and their corresponding z-scores.

The half-normal plot is obtained by plotting the ordered absolute values against their corresponding z-scores. As intended half-normal plot reveals outliers in the original sample, that appear only at the upper end of the plot. Except for these outliers, the rest of the distribution may appear to be normal.

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