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Guided Notes for Lesson: What is Normality Preliminaries Slide 1: Discussion Question: How do we describe data in Statistics? In this course, the way we describe data is by using the C.U.S.S. Method. We look at these characteristics: C. C U. U S. S S. S Slide 2 • Recall that a is a standard way of displaying the distribution of data graphically using the . • EX 1: Make a box plot using the following distribution of numbers in your calculator. • 2, 3, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9 10, 10, 10, 10, 10, 10, 11, 11, 12, 14, 14, 14, 15, 16, 18, 18, 18, 18, 19, 19, 20, 20, 22, 22, 24, 24, 25, 26, 26, 27, 28, 28, 28, 33, 45, 50, 55, 66 • After you obtain the graph in your calculator draw it in your notes, write down the five number summary, and then use C.U.S.S to describe the distribution. Slide 3: Calculator Use 1. Go to button and push 2. Copy the data into list 3. Push . to get to the 4. Select . page. . 5. Select Type: (bottom left) 6. Select Xlist: . 7. Select . Slide 5 Recall that a is a distribution using different has a similar shape to a a of the data of a and . The distribution . Below is a visual representation of . Video Questions 1. How do label the axes of a Histogram? 2. Are the bars of a Histogram connected? 3. What is the range of a Histogram? Slide 7: • As more is added to a Histogram the more and more similar to a becomes . You can see the resemblance in the examples we covered and in the following picture. Slide 8: • To assume that a set of data follows a of the most important is one in . • When we think about the concept of of numbers that has , consider a list , and If you have LOTS of these types of numbers then they MIGHT follow a “ ” • Some examples of Normally distributed data are: . • Can you think of some examples based on the description given above? Write them in your notes. Slide 9: The Normality Trap • Not all data follows a • Data with . or may not be distributed • will be closer to a distribution than samples • Real life data is almost . Characteristics of a Normal Distribution • The , , and • The curve is crosses the all have the same value and . about the line that • The curve approaches, but never touches the away from the • The . under the curve is equal to • Almost all of the as you move . under the curve exists within standard deviations of the mean. • When data follows a normal distribution it is denoted where 𝜇 is the mean and 𝜎 is the standard deviation of the distribution. How to Draw a Normal Curve Ex: In your notes draw the following curve. The data is N(10, 5). The Empirical Rule • Recall the characteristics of the Normal distribution said that almost all of the area under the curve exists within three standard deviations of the mean • How do we know what the almost means? For this we have the A.K.A The • The Rule states that; • of the data is within 1 of the mean • of the data is within 2 of the mean • of the data is within 3 of the mean • Discussion Question: Where do you think this rule originated from? Estimating Areas • Complete this example in your notes with a partner. • Suppose the scores from your AP Stats exam are N(75, 5). Answer the following questions. 1. Sketch the distribution 2. What percentage of the scores is within 60 and 90? 3. What percentage is the 50th percentile? 4. What score would be the 16th percentile? Z-Scores • A is how many away from the a data point is . • Z-scores are used to find the under the by standardizing it. • The Standard Normal Curve is a Normal curve with a mean of and standard deviation . • The formula for the z-score is as follows: • Now to interpret this result we would say: A score of 82 in the test is . • If your z-score is negative then the data is deviations away from the mean. standard Using Z-scores • Suppose we said that the show size for males in the U.S. is N(9.5, 1.25). • How would you calculate the percentage of shoe sizes that are below 10? • We could use the to estimate this, but an quicker way is to use . • Let’s work through this example with our calculators • Using the z-score formula we get: 𝑧 = 10−9.5 1.5 = .3333 • Now on the calculator press 2nd vars, normalcdf( , ) and you get the answer. • If it were to ask you for the percentage above 10, you would press 2nd vars, normalcdf( , ). Now You Try • Using the previous scenario answer the following questions in your notes with a partner. • Find the percentage of shoe sizes above 8.5 • Find the percentage of shoe sizes between 8.5 and 9.5 • Find the percentage of show sizes below 9.25 • Draw and shade the Normal Curves representing these scenarios Slide 19: • Discussion Question: How could we use the Empirical Rule to check if the data follows a Normal Distribution? The steps to checking the data for Normality are as follows 1. Check the (68-95-99.7 percent of the data lie within this boundary) 2. Check the (The closer the is to the mean suggest that the data might be Normally distributed) 3. Draw a or in your calculator and check C.U.S.S. The more symmetrical the graph the more evidence that suggest that the data is Normally Distributed Great White Sharks Activity Below are the length of a sample of 44 Great white Sharks in feet. In your notes use the steps previously stated to check if the length of this sample of Great White Sharks is Normally Distributed? Write a Paragraph describing your findings. Discuss your findings with your partner.