Overview and Descriptive Statistics STAT 155 A density histogram has one

Overview and descriptive statistics stat 155 a

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Chapter 1. Overview and Descriptive Statistics STAT 155 Exercise 1.27The paper “Study on the Life Distribution of Microdrills” (J.of Engr. Manufacture, 2002: 301305) reported the following observations,listed in increasing order, on drill lifetime (number of holes that a drill ma-chines before it breaks) when holes were drilled in a certain brass alloy.11142023313639444750596165676871747678798184858991939699101104105105112118123136139141148158161168184206248263289322388513(a) Construct a relative frequency histogram based on the equal-widthclass intervals 0-<50, 50-<100, 100-<150,· · ·, and comment onfeatures of the histogram.(b) Construct a histogram of the natural logarithms of the lifetime obser-vations, and comment on interesting characteristics.(c) What proportion of the lifetime observations in this sample are lessthan 100? What proportion of the observations are at least 200? 14
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Chapter 1. Overview and Descriptive Statistics STAT 155 15
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Chapter 1. Overview and Descriptive Statistics STAT 155 1.3 Measures of Location mean median quartiles trimmed mean Mean – the arithmetic average of the data. Mean is calcu- lated as the sum of all observations divided by the number of observations. Sample mean x of observations x 1 , x 2 , x 3 , · · · , x n is: x = x 1 + x 2 + · · · + x n n = n i =1 x i n For reporting x , it is recommended to use a decimal accuracy of one digit more than the accuracy of the x i ’s. The arithmetic mean is the most widely used measure of cen- tral location. However, it is oversensitive to extreme/outlying values and must be used with caution. Median – the middle value. Sample median e x is the middle sorted observation. That is, we want a value such that half of the data is smaller than it and half is greater than it. Steps to find the sample median : 1. Rank the n observations from smallest to largest. 16
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Chapter 1. Overview and Descriptive Statistics STAT 155 2. Ifnis odd, median equals to the middle value,ex= (n+12)thordered value;Ifnis even, there are two middle values whose averageequals the median,ex= average of(n2)thand(n+12)thordered values.Unlike the mean, the median is insensitive to extreme/outlyingvalues.In many samples, the relationship between the arithmetic meanand the sample median can be used to assess the shape of adistribution.
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