Z-Scores and The Standard Normal Distribution Midterm Review Z-Scores and The Standard Normal Distribution

1st and 2nd Exams Next Week - Midterm Exam Schedule Morning Afternoon Monday Regular B Day Schedule – Review Day Tuesday 1st and 2nd Exams 3rd and 4th REVIEW Wednesday 5th and 6th Exams 7th and 8th REVIEW Thursday 3rd and 4th Exams No Classes Friday 7th and 8th Exams

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𝝁=𝟎 𝝈=1

For All Normal Distributions

So… for the first three standard deviations from the mean we know how to find the percentage of data included! If we want to know the percentage of the population above, below, or between any values … we can use the z-score! The z-score tells us the number of standard deviations between any value and the mean of the population.  http://www.mathsisfun.com/data/standard-normal-distribution-table.html

Standard Deviation of population Mean of population value 𝑧= 𝑥−𝜇 𝜎 z-score Standard Deviation of population

Example C What is the z-score of a value of 27, given a set mean of 24, and a standard deviation of 2? 𝒛= 𝒙−𝝁 𝝈 𝒛= (𝟐𝟕)−(𝟐𝟒) (𝟐) 𝒛= 𝟑 𝟐 𝒛=𝟏.𝟓

Example D What is the z-score of a value of 104.5, in a set with 𝜇=125 and 𝜎=6.2? 𝒛= 𝒙−𝝁 𝝈 𝒛= (𝟏𝟎𝟒.𝟓)−(𝟏𝟐𝟓) (𝟔.𝟐) 𝒛= −𝟐𝟎.𝟓 𝟔.𝟐 𝒛=−𝟑.𝟑𝟏

Quick Practice 4. What is the z-score of the price of a pair of skis that cost $247, if the mean ski price is $279, with a standard deviation of $16? 5. What is the z-score of a 5-scoop ice cream cone if the mean number of scoops is 3, with a standard deviation of 1 scoop? 6. What is the z-score of the weight of a cow that tips the scales at 825 lbs, if the mean weight for cows of her type is 1150 lbs, with a standard deviation of 77 lbs?

Tips for reading Z-Score Tables Z-score tables give us the probability that a value or any value less than it, will occur. Always start by converting to a % …. (multiply by 100%) To find the percentage of values greater than a z-score, subtract this % from 100%.

𝟕𝟎.𝟓𝟒% What is the probability of a z-score less than 0.54? ____________   What is the probability of a z-score greater than 0.54? ____________ What is the probability of a z-score less than -0.93? ____________ What is the probability of a z-score greater than -0.93? ____________ What is the probability of a z-score greater than 0.81? ____________ What is the probability of a z-score less than -0.81? ____________ 𝟐𝟗.𝟒𝟔% 𝟏𝟕.𝟔𝟐% 𝟖𝟐.𝟑𝟖% 𝟐𝟎.𝟗𝟎% 𝟐𝟎.𝟗𝟎%

Examples E. What is the probability that a value with a z-score less than 2.47 will occur in a normal distribution? F. What is the probability that a value with a z-score greater than 1.53 will occur in a normal distribution? G. What is the probability of a random selection being less than 3.65, given a normal distribution with 𝜇=5 𝑎𝑛𝑑 𝜎=2.2?