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# Chapt8 - Chapter 8 The standard deviation as a ruler and...

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78 Chapter 8: The standard deviation as a ruler and the normal distribution A density curve is a “mathematical model” that describes a set of data. The NORMAL DISTRIBUTION describes many different data sets, like: o Weights of people o IQ o Calorie consumption per day o Lengths of pregnancies Attributes of the Normal Curve: Notation: Symmetric: Mean= μ Standard Deviation= ! The entire area under the curve is 100%, or 1 Z-Score calculation: ! μ " = X Z where μ (population mean) and ! (population standard deviation) are given. **Note: In these examples, we are talking about a single observation (X) coming from an entire population with a mean ( μ ) and a standard deviation ( ! )

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79 The Standard Normal Curve: Standardizing is a technique to help us determine percentages, proportions, probabilities, or area under the curve for any set of data. The new values are unit-less. The requirement is that the data must be normally or approximately normally distributed . We will use TABLE Z from your text. When we standardize into Z-scores: o Shape of the distribution does not change o The center does change. The mean becomes 0. o The spread does change. The standard deviation becomes 1. The Empirical Rule states that for any normal or approximately normal distribution, approximate percentages or proportions under the curve can be estimated. It is often called the 68-95-99.7% Rule.
80 Example: Use the following information for questions 1-3 The scores on a standardized English placement exam are normally distributed with a mean of 100 and standard deviation 9. 1. Students scoring below 82 on the placement test must take a remedial English course. What percentage of students has to take the remedial English course? 2. Approximately what percentage of placement test scores will fall between 82 and 109? Notation: P (82 < X < 109) A. 68% B. 81.5% C. 83.85% D. 95% Answer: 3. The top 16% of students will be placed in an Honors English class. What score must a student get on the placement test in order to be placed in Honors English? Notation: P (X > x)=0.16 Practice Problems: 1. A student got her verbal GRE scores back. The student got a 580 and was in the lower 16 th percentile. The report said the standard deviation of the GRE scores was 48. What is the mean of the scores?

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81 Use the following information to answer questions 2-4: In 2004, Stat119 exam scores followed a normal distribution with a mean of 75 points. Unfortunately, the value of the standard deviation and all of the other data got lost, except one Z-score. This Z-score was -2, with a corresponding raw exam score of 60. 2. Find the standard deviation of that exam’s results ( show work ) Answer: 3. Assume now that the mean for the exam was 70 with a standard deviation of 7. Using the Empirical Rule, an estimate for the proportion of values between 49 and 77 is: A) 0.68 B) 0.8385 C) 0.975 D) 0.815 E) 0.95 Answer: 4. For the exam with mean 70 and standard deviation 7, what score would a student have to
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Chapt8 - Chapter 8 The standard deviation as a ruler and...

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