5.6: Application of the Normal Distribution
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- 58909
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Now that we’ve explored the normal distribution and how z-scores standardize values, we’re ready to apply these tools to real-world problems.
We’ll work through five example scenarios that involve different types of probability questions using the normal distribution. You may use a z-table, calculator, statistical software, or graphing technology to answer these.
Reminder: For all these problems, assume the data is approximately normally distributed unless stated otherwise.
Example 1: Area to the Left (Lower Tail)
The heights of adult women in the U.S. are approximately normally distributed with a mean of 64 inches and a standard deviation of 3 inches.
Question: What proportion of adult women are shorter than 60 inches?
- Convert to a z-score: \( z = \frac{60 - 64}{3} \)
- Use a z-table or technology to find \( P(Z < z) \)
Example 2: Area to the Right (Upper Tail)
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
Question: What percentage of people have an IQ above 120?
- Find the z-score for 120
- Find the area to the left (from the table), then subtract from 1
- \( P(X > 120) = 1 - P(Z < z) \)
Example 3: Area Between Two Values
A college class takes a final exam. The scores are normally distributed with a mean of 72 and standard deviation of 8.
Question: What proportion of students scored a C?
- Find the z-scores for both endpoints (70 and 80)
- Use the table to find \( P(Z < z_2) \) and \( P(Z < z_1) \)
- Subtract to find area between: \( P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1) \)
Example 4: Percentile to Value (Reverse Z)
The SAT Math test scores are normally distributed with a mean of 520 and standard deviation of 100.
Question: What score corresponds to the 90th percentile? Recall that a student in the 90th percentile would score higher than 90% of other students. In otherwords, we have a proportion of 0.90
- Look up the area closest to 0.9000 in the z-table.
- Take that z-score and plug into: \( x = \mu + z\sigma \)
Example 5: Comparing Two z-scores
Jordan earns an 88 on their statistics test. The class mean is 75 with a standard deviation of 6.
However, Jordan scores an 82 on their biology test, where the mean is 70 with a standard deviation of 8.
Question: In which class did Jordan score higher relative to their peers?
- Calculate each class's z-score.
- Interpret: the higher positive z-score is the one further above average.
Normal Distribution Check-In
Use the standard normal table or technology for these questions. Answers rounded to 4 decimal places.
1. Z-score
A student scores 84 on a science test. The mean is 75 and the standard deviation is 5. What is the z-score?
2. Area to the Left
What proportion of data falls below a z-score of 0.85?
3. Area to the Right
What proportion of values are greater than a z-score of 1.2?
4. Area Between
What proportion of values fall between z = -1.0 and z = 1.0?
5. Value from Percentile
SAT Math scores are normally distributed with mean = 500 and standard deviation = 100. What score corresponds to the 84th percentile?
Tip for Practice
Use a z-table or software to determine areas under the standard normal curve. The goal here is to understand the logic and patterns — not just get numbers. These tools will be used again when we talk about confidence intervals and hypothesis testing.