normal distributions. essential question: how do you find percents of data and probabilities of...
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Normal Distributions
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Essential Question: How do you find percents of data
and probabilities of events associated with normal
distributions?
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Normal Curves (68-95-99.7 Rule)
• 68% of the data fall within 1 standard deviation of the mean.
• 95% of the data fall within 2 standard deviation of the mean.
• 99.7% of the data fall within 3 standard deviation of the mean.
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Normal Curve’s Symmetry
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Finding Areas Under a Normal Curve
• Suppose the masses (in grams) of pennies minted in the United States after 1982 are normally distributed with a mean of 2.5g and a standard deviation of 0.02g.
Find the following:
Percent of pennies that have a mass between 2.46g and 2.54g.
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Finding Areas Under a Normal Curve
• Suppose the masses (in grams) of pennies minted in the United States after 1982 are normally distributed with a mean of 2.5g and a standard deviation of 0.02g.
Find the following: The probability that a randomly chosen penny has a mass greater than 2.52g.
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Reflect 2a.
Explain how you know that the area under the curve between and represents 13.5% of the data if you know that the percent of the data within of the mean is 68% and the percent of the data within 2of the mean is 95%.
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The Standard Normal Curve
Standard Normal Distribution has a mean of 0 and a standard deviation of 1.
A data value from a normal distribution with a mean and standard deviation can be standardized by finding its z-score
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The Standard Normal Curve
• Areas under the standard normal curve to the left of a given z-score have be computed and appear in the standard normal table.
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Using the Z-Score
𝑃 (𝑧≤1.3 )=.9032𝑜𝑟 90.32%
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Example
• Suppose the heights (in inches) of adult females in the United States are normally distributed with a mean of 63.8 inches and a standard deviation of 2.8 inches.
• Fine each of the following: – The percent of women who are no more than 65
inches tall. – The probability that a randomly chosen woman is
between 60 inches and 63 inches tall.
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The percent of women who are no more than 65 inches tall.
• Convert 65 to a z-score:
• “no more than” means: ___
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𝑃 (𝑧≤0.4 )
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Reflect 3a
• Using this result, you can find the percent of females who are at least 65 inches tall without needing the table. Find the percent and explain your reasoning.
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The probability that a randomly chosen woman is between 60 inches and 63 inches tall.
• Convert 60 to a z-score:
• Convert 63 to a z-score:
) = =
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𝑃 (𝑧≤−0.3 )−𝑃 (𝑧 ≤−1.4) =
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Reflect 3b
• How does the probability that a randomly chosen female has a height between 64.6 inches and 67.6 inches compare with your answer? Why?