EQ: What is a normal distribution?
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1 Unit 5 - Statistics What is the purpose EQ: What tools do we have to assess data? this unit? What vocab will I need? Vocabulary: normal distribution, standard, nonstandard, interquartile range, population mean, z-score, boxplot, histogram, empirical rule, observational study, experiment. Jul 18 1:49 PM Normal Distribution What is the purpose of this lesson? EQ: What is a normal distribution? Jul 18 2:24 PM 1
2 What is the Normal distribution? Vocabulary: Normal Distribution - this is one type of distribution of data. Visually, a normal distribution is symmetric, single peaked bell curve. In this case, the empirical rule applies. What is the Empirical Rule? The Empirical Rule : - also known as the rule. - approx. 68% of all observations fall within one standard deviation of the mean. - approx. 95% of all observations fall within two standard deviations of the mean - approx. 99.7% of all observations will fall within three standard deviations of the mean. Jul 18 2:34 PM A. The Normal Distribution Curve What does the normal distribution curve look like? Jul 18 2:37 PM 2
3 B. Using the Empirical Rule ( Rule) Ex1: A normal distribution has a mean of 27 and a standard deviation of 5. Find the probability that a randomly selected x-value from the distribution is in the interval of 17 and 37. Jul 18 2:54 PM Ex2: A normal distribution has a mean of x and a standard deviation of. Find the probability that a randomly selected x-value from the distribution is in the interval P(x < x + 2 ). Mar 4 10:40 PM 3
4 Ex.3. The distribution of heights of young women aged 18 to 24 is approximately normal with a mean (mu) of 64.5 inches and a SD (sigma) of 2.5 inches. a. Draw a Normal curve to represent this data, clearly showing the application of the empirical rule. b. What percent of woman are taller than 69.5 inches? c. Between what heights do the middle 95% of woman fall? d. What percent of woman are shorter than 62 inches? e. A height of 67 inches corresponds to what percentile Mar 4 10:29 PM Ex.4. Scores for a professional exam are normally distributed with a mean of 650 and a standard deviation of 100. a. What is the probability that a randomly selected test score is between 450 and 850? b. Out of 1000 randomly selected test scores, how many would you expect to be between 450 and 850? c. Out of 2300 randomly selected test scores, how many would you expect to be between 650 and 950? Oct 5 8:13 PM 4
5 Vocabulary: What is a Standard Normal Distributions - when data standard values are standardized and normally normal distributed, the mean will be 0 and the distribution? standard deviation will be 1. Jul 18 2:34 PM C. Using Z scores and the normal curve How do we use a - An area under the normal distribution curve gives the proportion/percentile rank of the standard observations in a distribution. normal - The total area under the curve is 1 (or 100%). distribution? - If we are trying to find the proportion of observations that lie in a certain range of values, we need to find the area under the curve. - This is done using a table or the calculator (note: the values from the table always give the area to the left of the z score.) Jul 18 2:37 PM 5
6 How do I solve a problem using the standard Normal distribution? D. Finding Normal Proportions/ Percentages: Steps to take to find a percentage/ proportion: 1. Identify what it is you are trying to find. 2. Draw a picture to show the area under the standard normal curve. 3. Calculate a z score with the information given in the Q. 4. Use the z score, table and the fact that the total area under the curve is 1 to determine the % you are looking for. Oct 5 8:13 PM 1. Find the proportion of observations from the standard normal distribution that are less than Find the proportion of observations from the standard normal distributions that are greater than Jul 18 2:54 PM 6
7 How do I find a value if given a proportion? E. Finding Data Values given a % Steps to take: 1. Find the given proportion in body of the table. 2. Read the corresponding z-score from the left column and the top row. 3. Then, "unstandardize" the observed value (find the data value using the z score equation). Oct 25 5:43 PM 1. Find the point z with 25% of the observations falling to t left of it. 2. Find the point z with 40% of the observations falling to th right of it. May 11 10:23 PM 7
8 Ex 5: We know that the distribution of heights of adult American males is approximately normal, with a mean height of 69 inches and a standard deviation of 2.5 inches. a) What % of men are at least 6 ft tall? b) What % of men are between 5'1" and 6' tall? c) How tall must a man be to be in the tallest 10% of all adult men? Oct 25 5:35 PM Example Ex 6: The army reports that the distribution of head circumference among soldiers is approximately Normal with mean 22.8 inches and standard deviation 1.1 inches. Helmets are mass-produced for all except the smallest 5% and the largest 5% of head sizes. Soldiers in the smallest 5% or largest 5% get custommade helmets. What head sizes get custom made helmets? Oct 25 5:44 PM 8
9 Nov 15 4:40 PM 9
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