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Alg 2 7.7 z scores.notebook
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March 19, 2018
7.7 z scoresObjective:Calculate and apply zscores. Apply the standard normal distribution and zscores
Bridgette Jordan of Sandoval, Illinois (as of 2014), a town about 86 miles east of St. Charles, is one of the shortest women in the world, standing at 27 inches.
Alg 2 7.7 z scores.notebook
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March 19, 2018
Robert Wadlow was born in Alton, Illinois and passed away in 1940, at age 22, with a height of 107.1 inches.
Obviously, Bridgette is shorter than most women and Robert was taller than most men —but whose height is more unusual, relatively speaking? That is, relative to other adults, who is taller? We’ll say that that women have a mean of 64 in. and a standard deviation of 2.5 in. and that the mean height of men is 69.5 in. with a standard deviation of 2.8 in.
To decide, we'll use what is called a standardized score or zscore.
Alg 2 7.7 z scores.notebook
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March 19, 2018
This is the formula for a standardized score, standardized value, or zscore:
The zscore tells us how many standard deviations an observation is above or below the mean & zscores have no units.
Positive zscores indicate that an observation is higher than the mean & negative zscores indicate that an observation is lower than the mean.The mean for zscores is 0 and the standard deviation is 1.
OK, let's compare their heights using zscores.
Bridgette: 27 inches Robert: 107.1 incheswomen: μ=64 in. & σ=2.5 in. men: μ=69.5 in. & σ=2.8 in.
B 14.8R13.4
Alg 2 7.7 z scores.notebook
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March 19, 2018
Use your own height to find your zscore.women: μ=64 in. & σ=2.5 in. men: μ=69.5 in. & σ=2.8 in.Plot and label your zscore on the number line, then plot and label Bridgette's and Robert's zscores.If someone is average height, what is his/her zscore?
Which is more unusual?Plot each zscore on the number line.
A January high temperature of 57°F in Anchorage, Alaska, where January has μ=21°F & σ=10°F.
A January high temperature of 57°F in Honolulu, Hawaii, where January has μ=80°F & σ=8°F.
Alg 2 7.7 z scores.notebook
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March 19, 2018
A student in 2009 whose ACT score was 30, if μ=21.1 & σ=5.1
A certain car tire lasts 48800 miles, if μ=40000 miles & σ=5000 miles.
Which is more unusual?Plot each zscore on the number line.
A distribution with this shape is called a Normal Distribution.Notice that the distribution is symmetric and bellshaped.Almost no data exactly Normal, but lots of data is approximately Normal.
Some things that might be described as approximately Normal:heights of men, heights of women, IQ scores, SAT scores, ACT scores, average weights of cases of a given brand of potato chips,errors in measurements, fasting blood glucose levels, the percent that land "tails" if we had lots of people each toss 1000 coins, and the time it takes popcorn kernels to pop.
Alg 2 7.7 z scores.notebook
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March 19, 2018
When zscores are Normally distributed, we say that the distribution isthe Standard Normal Distribution.
When we standardize (using the zscore formula) things like heights, IQ scores, SAT scores, ACT scores, etc., we can use the Standard Normal Distribution to describe those zscores.
For things like standardized heights, notice that most of the zscores are near the average, 0. As you move farther and farther from the mean of 0, the distribution shows that those zscore values happen less and less frequently.
Remember that a distribution shows all the values that can happen (along the horizontal axis) and how often they happen (piled up above the axis).
Alg 2 7.7 z scores.notebook
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March 19, 2018
Time to use algebra to work backward!Suppose that January high temperatures in Anchorage, Alaska have μ=21°F & σ=10°F. If a certain January day has a high temperature with a zscore equal to 1.5, how warm did it get that day?
Time to use more algebra!Suppose that a certain phone battery will stream movies for an average μ=6 hours. If a randomly selected battery of that type lasted 8 hours and has a zscore equal to 2.5, then what is the standard deviation, σ, of this kind of battery?
