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3-4 example10

# 3-4 example10 - Student Grady Sinlonton Colu'se 1-Iat11119...

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Unformatted text preview: Student: Grady Sinlonton Colu'se: 1\-Iat11119: Elementary Statistics - Spring 3010 - C‘RN: 49339 Instructor: Shawn Pan-'ini - 16 weeks Date: 3.-"'18.-"'10 Book: Triola: Elementary Statistics. Me Time: 3:00 P1\-I The lengths (in inches) of 18 bears that were measured are given below. Find the 5-nurnber summary and construct a bcxplot. Does the distribution of the lengths appear to be symmetric or does it appear to be skewed? 40.5 48 53 45 6? 46.5 35 43 56 64 51 58 69.5 38 54 61.5 50 59 The 5-number summary consists of the smallest and largest numbers in the data set, the ﬁrst quartile, the median, and the third quartile. Be sure to ﬁrst list the data in ascending order. The data in ascending order are shown below. 35 38 40.5 43 45 46.5 48 50 51 53 54 56 58 59 61.5 64 6'? 69.5 From the list we see that the smallest number in the data set is 35, and the largest number in the data set is 69.5. Find the ﬁrst quartile, Q 1. Remember that the ﬁrst quartile is the 25th percentile. Use the formula for the locator. L = where k is the percentile and n is the number of values. L— A 13 —45 {100% )_' Since L is not a whole number, change L by rounding it up to the next larger number. Q1 is the ﬁfth number, counting from the lowest. Q 1 = 45 Find the median, M. Remember that the median is the 50th percentile. L— i 18 —9 {100% )— Since L is a whole number, add the 9th and the 10th number and divide by 2. The ordered list is repeated below for reference. 35 38 40.5 43 45 46.5 48 50 51 53 54 56 58 59 61.5 64 67 69.5 51+53 =52 2 M: Now ﬁnd the third quartile, Q3. Remember that the third quartile is the 75th percentile. L—[ij 18 —135 _100( )_ ‘ Page 1 Student: Grady Silnonton Colu'se: 1\-Iat11119: ElenIentaiy Statistics - Spring 3010 - C‘RN: 49339 Instructor: Shawn Pan-'ini - 16 weeks Date: 3.-"'18.-"'10 Book: T1‘iola: ElenIentaiy Statistics. Me Time: 3:00 P1\-I Since L is not a whole number, change L by rounding it up to the next whole number. Q3 is the 14th number, counting from the lowest. Q3 = 59 Thus, the 5-number summary is 35, 45, 52, 59, and 69.5. Use the 5-number summary to construct the boxplot. There are two different types of boxplots, skeletal (or regular) and modiﬁed. If the data contains any outliers, then a modified boxplot is constructed, otherwise a skeletal boxplot is constructed. A data value is an outlier if it is above Q3 by an amount greater than 1.5(IQR) or below Q I by an amount greater than 1.5(IQR). First ﬁnd the interquartile range. IQR = Qs—QI = 59—45 =14 Now ﬁnd Q 3 +1.5(IQR) and Q. — 1.5(IQR). Q3 +1.5(IQR)= 59 +1.5(l4)= 80 01 - 1.5(IQR) =45 - l.5(l4)= 24 Since the smallest number in the data set, 35, is larger than 24, and the largest number in the data set, 69.5, is smaller than 80, there are no outliers. Since there are no outliers, construct a skeletal (or regular) boxplot. The steps to constructng this boxplot are given below. - Construct a scale with values that include the minimum and maximum data values. - Construct a box (rectangle) extending from Q1 to Q3, and draw a line in the box at the median value. - Draw lines extending outward from the box to the minimum and maximum data values. Using the 5—number summary and the steps given above, the boxplot of the data is constructed. l—I—l l_'_l_'_l_'—l_'_l_'—I 30 4O 50 60 70 80 Use the horizontal lines on each side of the boxplot and the position of the median to determine whether the shape of the distribution is skewed or is roughly symmetric. Page 3 Student: Grady Sinlonton Course: 1\-Iat11119: Elenlentaiy Statistics - Spring 3010 - C‘RN: 49339 Instructor: Shawn Pan-'ini - 16 weeks Date: 3.-"'18.-"'10 Book: T1‘iola: Elenlentaiy Statistics. Me Time: 3:00 P1\-I If the median is near the center of the box and each horizontal line is ofapproximately equal length, the distribution is roughly symmetric. If the median is to the left of the center of the box or the right line is substantially longer than the left line, the distribution is skewed right. If the median is to the right of the center of the box or the left line is substantially longer than the right line, the distribution is skewed left. Notice that in the boxplot of the data the horizontal lines are approximately equal in length and the median is near the center of the box. Therefore, the distribution of the lengths is roughly symmetric. Page 3 ...
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