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The z-score is a measure that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations away from the mean. When dealing with normally distributed data, the z-score calculation is crucial for understanding where a particular score lies within the distribution.

A z-score is calculated using the formula:
\[ z = \frac{{X - \mu}}{{\sigma}} \]
where:\begin{itemize}\item \(X\) is the value in the distribution\item \(\mu\) is the mean of the distribution\item \(\sigma\) is the standard deviation of the distribution\end{itemize}
By standardizing values, z-scores allow for the comparison of scores on different kinds of distributions, enabling a researcher or statistician to determine how unusual or typical a result is. In our exercise, to find the value of \(c\) such that \(P(X>c) = .10\), we needed the z-score corresponding to the right-tail probability of 0.10, enabling us to locate our desired value relative to the distribution's mean.

Standard deviation is a statistical measure of the spread or variability of a set of data. It quantifies how much the values in a dataset deviate from the mean (average) of the dataset. Low standard deviation indicates that the values tend to be close to the mean, while high standard deviation signifies that the values are spread out over a wider range. The formula to calculate standard deviation (\(\sigma\)) for a population is:
\[ \sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}} \]
where:\begin{itemize}\item \(X\) represents each value in the dataset\item \(\mu\) is the mean of the dataset\item \(N\) is the total number of values in the dataset\end{itemize}In the context of a normal distribution, the standard deviation is particularly meaningful. As it is part of the z-score formula, it connects individual scores with the overall distribution. In our example, the standard deviation was given as the square root of the variance, simplifying our calculation to identify the value of \(c\).

The cumulative distribution function (CDF) for a normal distribution represents the probability that a random variable \(X\), under a normal distribution, assumes a value less than or equal to a given number. It is a non-decreasing, right-continuous function that provides the area under the normal curve to the left of a specific value.

The mathematical definition can be expressed as:
\[ F(x) = P(X \leq x) \]
where:\begin{itemize}\item \(F(x)\) is the CDF\item \(X\) is the random variable\item \(x\) is the value up to which you are cumulating the probabilities\end{itemize}
The CDF is crucial for probability calculations in normal distributions because it helps to determine the probability that a value will fall within a particular range. Use of the CDF is demonstrated in step 2 of our exercise, where we used the inverse of the cumulative function to determine the z-score associated with the 90th percentile (due to the symmetry of the standard normal curve). This provided us with the critical value needed to solve for \(c\).

Inverse cumulative probability, in the context of a normal distribution, is used to find the value of the variable such that the probability of the variable being less than or equal to that value equals the given cumulative probability. It is essentially the reverse operation of the cumulative distribution function (CDF) and is also referred to as the quantile function or the percent point function (PPF). This inverse operation allows you to work backwards from a probability to find a corresponding score or cutoff point.

Mathematically, the inverse cumulative probability is defined as:
\[ F^{-1}(p) = x \]\begin{itemize}\item where \(F^{-1}\) is the inverse function of the CDF\item \(p\) is the cumulative probability\item \(x\) is the value of the random variable corresponding to \(p\)\end{itemize}
In practice, most statistical software and calculators provide functions to compute this inverse directly. In the problem we've discussed, the inverse cumulative probability function was used to calculate the z-score \(z_c\) that corresponds to the given tail probability of 0.10. Knowing this z-score was the key step in finding the specific cut-off value \(c\) for our normal random variable.