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Standard Normal Distribution
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Density Curve
A smooth curve that describes the overall pattern of the distribution• The total area under the curve = 1• The curve will never go below the x-axis• The area between two values is equal to the
probability that the event will occur between those values
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Uniform DistributionA continuous random variable has uniform distribution if its values are spread evenly over the range of possibilities. The graph of a uniform distribution results in a rectangular shape.
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Uniform DistributionKim has an interview right after class. If the class runs longer than 51.5 minutes, she will be late. Given the previous uniform distribution, find the probability that a randomly selected class will last longer than 51.5 minutes.There is a 25% Chance the class will last longer than 51.5 minutes.
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Uniform Distribution
Find the probability that a time greater than 50.5 minutes is selected.
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Uniform Distribution
Find the probability that a time between 51.5 minutes and 51.6 minutes is selected.
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S tandard Normal DistributionIf a continuous random variable has a distribution with a graph that is symmetric and bell-shaped , as below, and it can be described by the following formula , then we say it has a normal distribution.
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Standard Normal DistributionThe standard normal distribution is a normal probability distribution with The total area under its density curve is equal to 1.
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Standard Normal DistributionThe standard normal distribution is a normal probability distributions with The total area under its density curve is equal to 1. All x-values are z-scores. A z-score refers to how many standard deviations above or below the mean a particular value is.The area under the curve is equal to its probability• Use Table A-2 (back cover of text) This table gives the area underneath the
curve to the left of the z-score.• Use Ti-83/84 [2nd][Vars] [2: normal cdf(] enter two z scores separate by a comma.
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Standard Normal DistributionThe precision Scientific Instrument company manufactures thermometers that are suppose to give readings of 0° at the freezing point of water. Tests on a large sample reveal that at the freezing point of water the readings of the thermometers are normally distributed with a mean of 0° and a standard deviation of 1.00°C. A thermometer is randomly selected. Find the probability of getting a reading less than 1.75.
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Standard Normal DistributionAssume that the readings on the thermometers are normally distributed with a mean of 0° and a standard deviation of 1.00°C. A thermometer is randomly selected. Find the probability of getting a reading greater than 1.96
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Standard Normal DistributionAssume that the readings on the thermometers are normally distributed with a mean of 0° and a standard deviation of 1.00°C. A thermometer is randomly selected. Find the probability of getting a reading less than .
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Standard Normal DistributionAssume that the readings on the thermometers are normally distributed with a mean of 0° and a standard deviation of 1.00°C. A thermometer is randomly selected. Find the probability of getting a reading between and .
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Standard Normal DistributionFind the indicated area under the curve of the standard normal distribution. About ____% of the area is between z = -2 and z = 2 (or within 2 standard deviations of the mean)
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Standard Normal DistributionFinding z scores from Known Areas1. Draw a bell-shaped curve and identify the region under the curve that corresponds to the given probability. If that region is not a cumulative from the left, work instead with a know region that is cumulative from the left.2. Using the cumulative area from the left,
– locate the closest probability using table A-2. – To find a z score corresponding using Ti-83/84 [2nd ][Vars][invNorm.] enter area cumulative from the left.
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Standard Normal DistributionAssume that the readings on the thermometers are normally distributed with a mean of 0° Find the temperature corresponding to the 95th percentile. That is find the temperature separating the top 5% from the bottom 95%.
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Standard Normal DistributionAssume that the readings on the thermometers are normally distributed with a mean of 0° Find the temperatures separating the bottom 2.5% and the top 2.5%.
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Homework!
• 6.1 : 6- 39 eoo, 41- 49 odd.