1 / 14

Chapter 5 Standard Deviation as a Ruler and Normal Models

Chapter 5 Standard Deviation as a Ruler and Normal Models. In this chapter, we will look at using the standard deviation as a measuring stick and some properties of data sets that are normally distributed.

alden-farmer
Download Presentation

Chapter 5 Standard Deviation as a Ruler and Normal Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 5Standard Deviation as a Rulerand Normal Models In this chapter, we will look at using the standard deviation as a measuring stick and some properties of data sets that are normally distributed.

  2. The number of standard deviations a particular observation in a data set is away from the mean of that set is called it z-score, and is denoted zx. It is calculated as follows: z-ScoresDefinition & Formula

  3. If zx > 0, then the observation is above average If zx < 0, then the observation is below average z-scores have no units; when observations are changed into their z-scores it is called standardizing the data Once standardized, observations from different data sets having different centers and spreads can be compared. z-ScoresConceptual Ideas

  4. In the year 2009, cars sold in the United States had an average of 135 horsepower with a standard deviation of 40 horsepower. (a) Find the z-score for a car with 120 horsepower. (b) Find the z-score for a car with 200 horsepower. (c) A certain car has z-score of 1.2. How much horsepower does it have? (d) A certain car has z-score of 0.6. How much horsepower does it have? Example 1

  5. Adult Dalmatians have an average weight of 52 pounds with a standard deviation of 6 pounds. Adult German Shepherds have an average weight of 77 pounds with a standard deviation of 3.6 pounds. Which is more unusual, an adult Dalmatian weighing 63 pounds or an adult German Shepherd weighing only 68 pounds? Example 2

  6. If the histogram of a data set is unimodal and fairly symmetric, then its distribution is called normal. If such a distribution has mean m and standard deviation , it can be modeled with a normal model, the notation of which is , and the shape is below: Normal ModelsDefinition and Ideas  + 3   + 2  +   -   - 2  - 3

  7. The standard normal curveis . This is commonly called the z–curve. It has shape below: Normal ModelsDefinition and Ideas 3 2 1 -3 -1 0 -2

  8. For a normal distribution: ≈ 68% of values fall within one standard deviation of the mean ≈ 95% of values fall within two standard deviations of the mean ≈ 99.7% of values fall within three standard deviations of the mean Normal Models68 – 95 – 99.7 Rule

  9. IQ scores are modeled by . • Draw the 68-95-99.7 diagram for IQ scores. This should be used for the remaining parts of this problem. • About what percent of people have IQ scores 68 and 84? • About what percent of people have IQ scores above 116? • About what percent of people have IQ scores below 68? • What is the 84th percentile of IQ scores? That is, about what score can we say about 84% of people get that score or lower? Example 3

  10. For a normal distribution, the 68-95-99.7 diagram is somewhat limited in terms of what questions can be answered. For example, in the previous example, if we wanted to know what percentage of people have IQ scores between 90 and 120, the 68-95-99.7 diagram would not be very helpful. For such questions we can use z-scores and tables published in statistics books or we can use technology. We will focus on technology. Normal ModelsMore General Use

  11. gives the proportion of data that falls between the values a and b from a normal model with mean  and standard deviation . • if there is no lower bound, the a can be replaced with -∞ • if there is no upper bound, the b can be replaces with ∞ To find this command press on the TI-83/84. Normal ModelsTechnology

  12. Birth weights of babies are well modeled by . • About what percent of babies are born between 3200g and 3600g? • About what percent of babies are born weighing more than 4000g? • About what percent of babies are born weighing less than 2000g? Example 4

  13. We can also find percentiles using the TI 83/84. gives the value on the normal curve such that (100a)% of the data is less than or equal to this value. This command is found by pressing Normal ModelsMore Technology

  14. Birth weights of babies are well modeled by . • Find the 15th percentile of birth weights. • Find the lowest 5% of all birth weights. • Find the highest 2% of all birth weights. Example 5

More Related