5.4: Finding Critical Values from the Normal Distribution
- Page ID
- 48844
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Sometimes we know the proportion or area that is defined by a range of normally distributed values. But we may not know the cutoff values (also called critical values) for the range.
Suppose we want to find the critical value that separates the top 15% of men’s heights from the lower 85%. We can use technology to calculate this value from the standard normal distribution. This value is a Z-score.
Using desmos to Find a Critical Value from The Standard Normal Distribution
Suppose we want to find the critical value that separates the top 15% of men’s heights from the lower 85%. We can use technology to calculate this value from the standard normal distribution. This value is a Z-score.
Images are created with the graphing calculator, used with permission from Desmos Studio PBC.
To calculate a value of z from a given probability, use the following steps:
- Go to https://www.desmos.com/calculator.
- The function we will use takes the area or probability that is less than the critical value. Let’s call this area A. It returns a value of \(z\). In the first line, type normaldist().inversecdf(A).
In our example above, we learn that \(z= \text{normaldist}().\text{inversecdf}(0.85)\approx1.036\) is the Z-score that separates the lower 85% from the upper 15%. This is useful information, but we haven’t yet found the adult male height that separates the lower 85% from the upper 15%. We need to find the adult male height that corresponds to the Z-score 1.036.
Recall, that a Z-score represents the distance (in standard deviations) a value is above or below the mean. Therefore, the male height that corresponds to this Z-score is 1.036 standard deviations above the mean (since it is positive). \(x\) represents random adult male heights, the mean of the distribution is 69 inches, and the standard deviation of the distribution is 3 inches. Let’s translate this into a mathematical sentence:
\[\begin{aligned}\underline{\text {This adult male height}}\text{ is } 1.036\ \underline{\text{standard deviations}}\text{ above }\underline{\text{the mean}}\\
\quad\quad\quad\quad\quad\quad\quad x \quad\quad\quad\quad\quad = 1.036 \quad\quad\quad\quad\quad\quad \sigma \quad\quad\quad + \quad\quad\quad \mu\quad
\end{aligned}\]
\[x=1.036 \cdot 3+69=72.108 \text { inches }\nonumber\]
Therefore, the adult male height that separates the lower 85% from the top 15% is 72.108 inches. Furthermore, we can say that this adult male height is in the 85th percentile because said adult man is taller than 85% of all adult men.
In general, to convert a Z-score to an x-value that is from a normal population that has mean and standard deviation , we use the formula
\[x=z \cdot \sigma+\mu\nonumber\]
You try!
- The Welcher Adult Intelligence Test Scale is composed of a number of subtests. On one subtest, the raw scores have a mean of 35 and a standard deviation of 6. Assuming these raw scores form a normal distribution:
- What number represents the 65th percentile (what number separates the lower 65% of the distribution)?
- What number represents the 90th percentile?
- What numbers separate the middle 95% of scores?
- What number represents the 65th percentile (what number separates the lower 65% of the distribution)?
- Kelly’s score on the SAT was in the 92nd percentile. Explain what this means about her score relative to all students who took the SAT.