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# Applied 40S April 6, 2009

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Introduction to working with the Standard Normal Curve.

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### Applied 40S April 6, 2009

1. 1. Working with z-scores z by Shamana ©
2. 2. The data below shows the ages in years of 30 trees in an area of natural vegetation. 37 15 34 26 25 38 19 22 21 28 42 18 27 32 19 17 29 28 24 35 35 20 23 36 21 39 16 40 18 41 Determine whether the data approximate the normal distribution. USING the 68 -95-97 RULE
3. 3. HOMEWORK The following are the number of steak dinners served on 50 consecutive Sundays at a restaurant. 41 52 46 42 46 36 44 68 58 44 49 48 48 65 52 50 45 72 45 43 47 49 57 44 48 49 45 47 48 43 45 56 61 54 51 47 42 53 44 45 58 55 43 63 38 42 43 46 49 47 Draw a suitable histogram that has ﬁve bars.
4. 4. HOMEWORK The frequency table shows the ages of all the students in Senior 4 Math at Newberry High. Find the mean, μ. Then calculate the percent of students older than the mean age. How does this compare to the percent of students older than the mean age if the distribution were a normal distribution? Based on this answer, does it seem that the students' ages approximate a normal distribution? Age of Student 15 16 17 18 19 20 21 22 # of Students 1 7 42 24 7 4 2 1
5. 5. Now let's try a problem involving Grouped Data A machine is used to ﬁll bags with beans. The machine is set to add 10 kilograms of beans to each bag. The table shows the weights of 277 bags that were randomly selected. wt in kg 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 # of bags 1 3 13 25 41 66 52 41 25 7 3 (a) Are the weights normally distributed? How do you know? (b) Do you think that using the machine is acceptable and fair to the customers? Explain your reasoning.
6. 6. Properties of a Normal Distribution The shape of any normal distribution curve is determined by: • the mean (μ) • the standard deviation (σ) Changing the mean will shift the graph horizontally. Changing the standard deviation will change the shape of the curve, making it narrow or wide.
7. 7. Properties of a Normal Distribution The data are continuous and distributed evenly around the mean, and the graph created by the data is a bell-shaped curve, as shown in the examples below. These curves represent data sets that have the same mean, but different standard deviations. Which one has a larger standard deviation (σ)? How can you tell?
8. 8. The contents in the cans of several cases of soft drinks were tested. The mean contents per can is 356 mL, and the standard deviation is 1.5 mL. (a) Two cans were randomly selected and tested. One can held 358 mL, and the other can 352 mL. Calculate the z-score of each. (b) Two other cans had z-scores of -3 and 1.85. How many mL did each contain?
9. 9. Curving The Marks Professor Adams has 140 students who wrote a statistics test. If the marks are approximately normally distributed: • how many students should have a 'B' mark (i.e., 70 to 79 percent)? • how many students should have failed (i.e., less than 50 percent)? • how high must she set the mark for an 'A' if she wants 5 percent of the students to get an A? • how high must she set the passing mark if she wants only the top 75 percent of the marks to be passing marks?