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Statistics Chapter 2 Exploring Distributions

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Section 2.5 The Normal Distribution. Statistics Chapter 2 Exploring Distributions. Central Intervals for Normal Dist. 68% of values lie within 1 SD of the mean. Including to the right and left 90% of the values lie with 1.645 SDs of the mean.

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central intervals for normal dist
Central Intervals for Normal Dist.
  • 68% of values lie within 1 SD of the mean.
    • Including to the right and left
  • 90% of the values lie with 1.645 SDs of the mean.
  • 95% lie within about 2 SDs (actually 1.96 SDs) of the mean.
  • 99.7% of the data lie within 3 SDs of the mean.
why do we study the normal distribution
Why do we study the Normal Distribution?
  • Very common distribution of data throughout many disciplines.
    • SAT / ACT scores
    • Measure of diameter of tennis balls
    • Heights / weights of people
  • Once we know a distribution is Normal, there is a tremendous amount of information we can determine or predict about it.
normal distribution
Normal Distribution
  • All normal distributions have the same basic shape.
    • The difference: tall and thin vs. short and fat
    • However, we could easily stretch the scale of the tall thin curve to make it identical to the short fat one.
  • The area under the curve can be thought of in terms of proportions or percentage of data.
    • The total area under the curve is 1.0 (100%)
the standard normal distribution
The Standard Normal Distribution
  • We can standardize any normal curve to be identical.
  • We do this by treating the mean as Zero and the SD as One.
  • The variable along the x-axis becomes what we call a z score.
  • The z score is the number of SDs away from the mean.
finding z scores for the standard normal distribution
Finding z scores for the Standard Normal Distribution
  • Practice problems:
  • 1) Normal distribution with:
    • mean = 45 and SD = 5
    • Find the z score for a data value of 19
    • Find the z score for a data value of 52
  • 2) Normal distribution with:
    • Mean = 212 and SD = 24
    • Find the z score for a data value of 236
proportion of data in a range
Proportion of data in a range
  • We can use the standard normal curve to find proportion of data in a range of values.
  • Normal Curve example: SAT I Math scores
    • Mean = 500 SD = 40
    • Find the proportion of data in the score range 575 or less.
  • Using z tables: Table A very back of book
    • Find the proportion of data above 575.
    • Find the proportion of data between 490 and 550.
  • Read all of 2.5.
  • Be prepared for quiz on Tuesday.
2 5 quiz
2.5 Quiz
  • You have collected data regarding the weights of boys in a local middle school. The distribution is roughly normal. The mean is 113 lbs and the SD is 10 lbs.
    • A) What proportion of boys are below 100 lbs?
    • B) What proportion are above 120 lbs?
    • C) What proportion are in between 90 & 120 lbs?
using calculator for proportions
Using calculator for proportions
  • You can also use the TI-83 or higher to find these same proportions:
  • 2nd , Distr, normalcdf(low, high,mean,SD)
  • When using z scores you can leave mean,SD blank. normalcdf(low, high) it will default to mean=0 and SD=1.
  • This will give you the same area under the curve (proportion of data) as the z table.
finding the z score from the percent
Finding the z score from the Percent
  • If you know the percent of data covered under a normal distribution, you can find the z-score.
  • Simply look up the percent (proportion) in the z table and relate it to the corresponding z score.
    • Find the value that is closest to the percent given
  • Another method is with the calculator.
    • 2nd ,Distr, invNorm(proportion, mean, SD)
  • Find the z-score that has the given percent of values below it in a standard normal distribution:
    • a) 32% b) 41% (use the z-table)
    • c) 87% d) 94% (use your calculator)
using the z score to find a value
Using the z-score to find a value
  • If you know how many SDs a value is from the mean, you can use this (z-score) to find the actual data value:
    • x = mean + (z • SD)
  • Example: The mean weight of the boys at a middle school is 113 lbs, with a SD of 10 lbs. One boy is determined to be 2.2 SDs above the mean. How much did the boy weigh?
combining the last two situations
Combining the last two situations
  • So now, if you know the percentage of data above or below a data value and you know the mean and SD, you can figure out that data value:
    • Use z-table to find the z-score, then use the z score with mean and SD to find the data value.
    • Or you can use the invNorm function on your calc.
      • invNorm(proportion, mean, SD)
  • The heights of U.S. 18-24 yr old females is roughly normally distributed with a mean of 64.8 in. and a SD of 2.5 in.
    • Estimate the percent of women above 5’8”
    • What height would a US female be if she was 1.5 SDs below the mean? Give your answer in ft & in.
    • What height would a US female be if she was considered to be in the 80th percentile?
review examples
Review Examples
  • What percentage of US females is above 5’7”?
  • What percent are between 5’7” and 5’0”?
trick question
Trick Question
  • The cars in Clunkerville have a mean age of 12 years and a SD of 8 years. What percentage of cars are more than 4 years old?
    • Why is this a trick question?
  • Page 93
    • E59, 61, 63, 64, 67, 69, 71, 73, 74