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Statistics Chapter 2 Exploring Distributions

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Section 2.5

The Normal Distribution

StatisticsChapter 2 Exploring Distributions

- 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.

- 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.

- 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%)

- 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.

- 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

- 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.

- 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?

- 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.

- 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)

- 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?

- 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?

- What percentage of US females is above 5’7”?
- What percent are between 5’7” and 5’0”?

- 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