## 3.45

**Part (a)**

The key is in chapter 2, where it explains that the standard deviation is measured in the same units as the data presented in the experiments.

For instance, in the normal distribution ,**2.5 standard deviations below the mean**will be at .

In the particular case of the standard normal distribution , the -score is at .

Now, we only need to look up Table A for the -score –2.5. Notice that the corresponding percentage is 0.00621 (that is our 0.6%)**Part (b)**

According to the wording of part (b), older women have a mean BMD “two units smaller than that of young adults”; therefore, as the standard deviation is also the same, we can think of the normal distribution of BMD for women as .

In this case, the*z*-score that indicates the percentage of older women with osteoporosis will be . We look up Table A for –0.5, and the corresponding percentage is 0.3085 (that is, 30.85%)

### Leave a Reply Cancel reply

### We have moved!

On November 2014, I migrated this blog to blancosilva.github.io. Please update your bookmarks and RSS feeds accordingly.

### In the news:

### Recent Posts

- Migration
- Computational Geometry in Python
- Searching (again!?) for the SS Central America
- Jotto (5-letter Mastermind) in the NAO robot
- Robot stories
- Advanced Problem #18
- Book presentation at the USC Python Users Group
- Areas of Mathematics
- More on Lindenmayer Systems
- Some results related to the Feuerbach Point
- An Automatic Geometric Proof
- Sympy should suffice
- A nice application of Fatou’s Lemma
- Have a child, plant a tree, write a book
- Project Euler with Julia
- Seked
- Nezumi San
- Ruthless Thieves Stealing a Roll of Cloth
- Which one is the fake?
- Stones, balances, matrices
- Buy my book!
- Trigonometry
- Naïve Bayes
- Math still not the answer
- Sometimes Math is not the answer
- What if?
- Edge detection: The Convolution Approach
- OpArt
- So you want to be an Applied Mathematician
- Smallest Groups with Two Eyes
- The ultimate metapuzzle
- Where are the powers of two?
- Geolocation
- Boundary operators
- The Cantor Pairing Function
- El País’ weekly challenge
- Math Genealogy Project
- Basic Statistics in sage
- A Homework on the Web System
- Apollonian gaskets and circle inversion fractals
- Toying with basic fractals
- Unusual dice
- Wavelets in sage
- Edge detection: The Scale Space Theory
- Bertrand Paradox
- Voronoi mosaics
- Image Processing with numpy, scipy and matplotlibs in sage
- Super-Resolution Micrograph Reconstruction by Nonlocal-Means Applied to HAADF-STEM
- The Nonlocal-means Algorithm
- The hunt for a Bellman Function.
- Presentation: Hilbert Transform Pairs of Wavelets
- Presentation: The Dual-Tree Complex Wavelet Transform
- Presentation: Curvelets and Approximation Theory
- Poster: Curvelets vs. Wavelets (Mathematical Models of Natural Images)
- Wavelet Coefficients
- Modeling the Impact of Ebola and Bushmeat Hunting on Western Lowland Gorillas
- Triangulations
- Mechanical Geometry Theorem Proving

### Pages

- About me
- Books
- Curriculum Vitae
- Research
- Teaching
- Mathematical Imaging
- Introduction to the Theory of Distributions
- An Introduction to Algebraic Topology
- The Basic Practice of Statistics
- MA598R: Measure Theory
- MA122—Fall 2014
- MA141—Fall 2014
- MA142—Summer II 2012
- MA241—Spring 2014
- MA242—Fall 2013
- Past Sections
- MA122—Spring 2012
- MA122—Spring 2013
- Lesson Plan—section 007
- Lesson Plan—section 008
- Review for First part (section 007)
- Review for First part (section 008)
- Review for Second part (section 007)
- Review for Third part (section 007)
- Review for the Second part (section 008)
- Review for the Fourth part (section 007)
- Review for Third and Fourth parts (section 008)

- MA122—Fall 2013
- MA141—Spring 2010
- MA141—Fall 2012
- MA141—Spring 2013
- MA141—Fall 2013
- MA141—Spring 2014
- MA141—Summer 2014
- MA142—Fall 2011
- MA142—Spring 2012
- MA241—Fall 2011
- MA241—Fall 2012
- MA241—Spring 2013
- MA242—Fall 2012
- MA242—Spring 2012
- First Midterm Practice Test
- Second Midterm-Practice Test
- Third Midterm—Practice Test
- Review for the fourth part of the course
- Blake Rollins’ code in Java
- Ronen Rappaport’s project: messing with strings
- Sam Somani’s project: Understanding Black-Scholes
- Christina Papadimitriou’s project: Diffusion and Reaction in Catalysts

- Problem Solving
- Borsuk-Ulam and Fixed Point Theorems
- The Cantor Set
- The Jordan Curve Theorem
- My oldest plays the piano!
- How many hands did Ernie shake?
- A geometric fallacy
- What is the next number?
- Remainders
- Probability and Divisibility by 11
- Convex triangle-square polygons
- Thieves!
- Metapuzzles
- What day of the week?
- Exact Expression
- Chess puzzles
- Points on a plane
- Sequence of right triangles
- Sums of terms from Fibonacci
- Alleys
- Arithmetic Expressions
- Three circles
- Pick a point
- Bertrand Paradox
- Unusual dice
- El País’ weekly challenge
- Project Euler with Julia

- LaTeX

### Categories

aldebaran
algebra
algorithm
analysis
applied mathematics
approximation theory
books
calculus
catalyst
chemical engineering
circumcenter
coding
combinatorics
computational geometry
curvelets
data mining
Delaunay
denoising
differential geometry
edge detection
engineering
Euclidean geometry
fractal
functional analysis
geometry
harmonic analysis
history of math
image
image processing
imdb
Japan
L-system
LaTeX
latitude
Lindenmayer system
linear algebra
longitude
mathematical imaging
mathematics
matplotlibs
Measure Theory
metacritic
movie critics
movies
nonlocal means
numerical analysis
numpy
oxide
pattern recognition
physics
probability
programming
puzzles
python
sage
scanning transmission electron microscopy
scientific computing
scipy
segmentation
signal processing
statistics
stats
STEM
super-resolution
tex
tikz
tomography
topology
triangle
triangulation
trigonometry
Voronoi
voronoi diagram
wavelets
weights

### Archives

- November 2014
- September 2014
- August 2014
- July 2014
- June 2014
- March 2014
- December 2013
- October 2013
- September 2013
- July 2013
- June 2013
- April 2013
- January 2013
- December 2012
- August 2012
- July 2012
- June 2012
- May 2012
- April 2012
- November 2011
- September 2011
- August 2011
- June 2011
- May 2011
- April 2011
- February 2011
- January 2011
- December 2010
- May 2010
- April 2010
- September 2008
- September 2007
- August 2007

### @eseprimo

Error: Twitter did not respond. Please wait a few minutes and refresh this page.

### Math updates on arXiv.org

- Second-order Bounds of Gaussian Kernel-based Functions and its Application to Nonlinear Optimal Control with Stability. (arXiv:1707.06240v1 [math.OC])
- Rigorous free fermion entanglement renormalization from wavelet theory. (arXiv:1707.06243v1 [quant-ph])
- Multi-point correlations for two dimensional coalescing random walks. (arXiv:1707.06250v1 [math.PR])
- Measurable process selection theorem and non-autonomous inclusions. (arXiv:1707.06251v1 [math.DS])
- Some Results on Joint Record Events. (arXiv:1707.06254v1 [math.PR])
- Zero-temperature limit of quantum weighted Hurwitz numbers. (arXiv:1707.06259v1 [math-ph])
- A Result on Relative Conormal Spaces. (arXiv:1707.06266v1 [math.AG])
- On Newstead's Mayer-Vietoris argument in characteristic 2. (arXiv:1707.06268v1 [math.GT])
- On functional tightness of infinite products. (arXiv:1707.06269v1 [math.GN])
- Induced Good Gradings of Structural Matrix Rings. (arXiv:1707.06270v1 [math.RA])

### sagemath

- An error has occurred; the feed is probably down. Try again later.

Thank you so much!

Fransisco, thank you. But I am a little confused. I understand the reasoning, but isn’t the formula for z score x – mean divided by deviation. However, you subtracted the mean from the x, the opposite from the formula? Can you please explain why you are subtracting the mean from x, instead of using the formula to subtract the x from the mean?? I am confused why sometimes, in the earlier exaples, we subtracted x from mean, but here in this problem you are subtracting the mean from x? Can you please explain? Thank you…

Not exactly: the formula that you mention helps you find the z-score in a situation where the mean is not zero and/or the std is not one: they will give you the , the values of and , and then you apply that formula to find the z-score.

Notice that for example, in the first part, they are telling us directly what the z-score is:

2.5 standard deviations below the mean.Since they are offering your neither , nor mean () nor std (), we are playing directly with the , so no adjustment is necessary. The trick there was to realize that the standard deviation is measured in the same units as the variables are measured.

Fransisco-

I have been studying, and I am a little confused and was wondering if you could clarify something for me – how do we know the z-scores for the 68-95-99.7 rule? I mean, what z-scores reflect these 3 percentages undert the cuve? What z-scores correspond to the 68-95-99.7 rule? Would you mind helping me on that? I am sorry, but I am confused on this!

Scott

The best way to go about it, in my opinion, is by visualization of the graph of the normal distribution, pointing up where the cuts for the 68%, 95% and 99.7% are.

The image below is a generic normal distribution .

In the case of the standard normal distribution and , the corresponding scores would be:

For 50%: 0

For (50 + 34)%: 1.

For (50 – 34)%: -1

For (50 + 34 + 13.6)%: 2.

For (50 – 34 – 13.6)%: -2

and so on…

Thank you so much Fransisco – this really helped me understand this concept! I appreciate all your help