Section 3:4 Answers page 173 #5,6, 12, 14, 16, 19, 21 23, 28 #5. To qualify for Mensa, one needs to have an IQ that is in the top 2% of people #6. Comparing z-score allows for a unitless comparison of the # of standard deviations. This allows us to have a standard basis for comparison. #12. 67-inch man: z = -0.87 62-inch woman: z = -0.55 The 62-inch woman is relatively taller than the 67-inch man #14. Ted Williams: z=3.82 Magglio Ordonez: z = 2.86 Ted Williams had the better year relative to his peers

Section 3:4 Answers page 173 #5,6, 12, 14, 16, 19, 21 23, 28 #16. Within 2 s.d. (+/2) z = -2 = (x-8)/0.05 z = 2 = (x-8)/0.05 x = 7.9 x = 8.1 Any bolts less than 7.9 cm or greater than 8.1 cm in length will be destroyed. #19 a.) (answers vary) b.) IQR = 256.9 crimes / 100,000 This means that the middle 50% of all observations have a range of 256.9 crimes / 100,000 c.) LF = -112.55 UF = 915.05 Since 1,459 is above the upper fence, the Washington DC crime rate is an outlier.

Section 3:4 Answers page 173 #5,6, 12, 14, 16, 19, 21 23, 28 #21. a.) z = -1.70 The 1971 rainfall amount is 1.70 standard deviations below the mean b.) Q1=2.625 inches, Q2 = M = 3.985 inches Q3 = 5.36 inches c.) IQR = 2.735 inches The range of the middle 50% of the observations of Chicago rainfall in April is 2.735 inches d.) LF = -1.478 inches UF = 9.463 inches There are no outliers #23. a.) Q1 = -.02 Q2 = 0.035 Q3 = 0.095 b.) LF = -0.1925 UF = 0.2675 The return 0.47 is an outlier because it is great than the UF

Section 3:4 Answers page 173 #5,6, 12, 14, 16, 19, 21 23, 28 #28.) a.) LF = -$28.5 UF = $103.5 $115 and $1,000 are outliers because they are both greater than the Upper Fence b.) c.) (Answers vary)

Chapter 3 Numerically Summarizing Data Section 3.5 Five Number Summary; Boxplots

Objective(s) To compute the five-number summary To draw and interpret boxplots

Warm-up Type 1 Writing / 3 lines or more – 2 minutes How many quartiles are there? What do Q1, Q2, Q3 represent? What is Q4? What do you think the 5-number summary means?  

We need the maximum and minimum to determine if outliers exist. Remember Q1, Q2, Q3 are “resistant” to extreme values. It does not give us any information about outliers. We need the maximum and minimum to determine if outliers exist. These 5 combined make up the 5-number summary  

The Five-Number Summary MINIMUM Q1 M Q3 MAXIMUM

EXAMPLE Finding the Five Number Summary Find the five number summary for the employment ratio data from Section 3.4.

Steps for Drawing a Boxplot Step 1: Determine the lower and upper fence: Lower Fence = Q1 - 1.5(IQR) Upper Fence = Q3 + 1.5(IQR)

Steps for Drawing a Boxplot Step 1: Determine the lower and upper fence: Lower Fence = Q1 - 1.5(IQR) Upper Fence = Q3 + 1.5(IQR) Step 2: Draw vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box.

Steps for Drawing a Boxplot Step 1: Determine the lower and upper fence: Lower Fence = Q1 - 1.5(IQR) Upper Fence = Q3 + 1.5(IQR) Step 2: Draw vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box. Step 3: Label the lower and upper fence.

Steps for Drawing a Boxplot Step 1: Determine the lower and upper fence: Lower Fence = Q1 - 1.5(IQR) Upper Fence = Q3 + 1.5(IQR) Step 2: Draw vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box. Step 3: Label the lower and upper fence. Step 4: Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence.

Steps for Drawing a Boxplot Step 1: Determine the lower and upper fence: Lower Fence = Q1 - 1.5(IQR) Upper Fence = Q3 + 1.5(IQR) Step 2: Draw vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box. Step 3: Label the lower and upper fence. Step 4: Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence. Step 5: Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk (*).

EXAMPLE Drawing a Boxplot Draw a boxplot for the employment ratio data.

Distribution Shape Based Upon Boxplot 1. If the median is near the center of the box and each of the horizontal lines are approximately equal length, then the distribution is roughly symmetric. 2. If the median is left of the center of the box and/or the right line is substantially longer than the left line, the distribution is right skewed. 3. If the median is right of the center of the box and/or the left line is substantially longer than the right line, the distribution is left skewed

Determine the shape of the distribution from the graph Symmetric Skewed Left Skewed Right

Symmetric

Determine the shape of the distribution from the graph Symmetric Skewed Left Skewed Right

Skewed Right

Determine the shape of the distribution from the graph Symmetric Skewed Left Skewed Right

Skewed Left

EXAMPLE Identify the Shape of a Distribution from a Boxplot Determine the shape of the employment ratio data based on the boxplot.

Determine the shape of the employment ratio data based on the boxplot Symmetric Skewed Left Skewed Right

EXAMPLE Comparing Two Data Sets Using Boxplots The following data represent the birth rate for women 15 - 44 years of age in 1990 and 1997 for each state. Draw boxplots for each year using the same scale.

60.5 63.9 68.0 66.1 64.8 74.2 85.3 64.1 65.9 66.1 64.7 67.3 67.4 86.3 60.5 66.5 73.8 56.1 72.9 65.9 79.2 62.4 73.0 69.1 65.5 70.1 77.7 63.6 69.4 70.3 69.7 72.1 82.3 64.2 64.7 68.9 69.7 67.5 92.0 69.1 66.3 67.2 71.5 77.2 77.6 67.0 64.8 71.7 62.8 65.9 68.7 1990: 49.5 68.4 60.3 61.2 64.2 55.9 61.8 52.4 60.0 58.9 62.1 70.4 60.7 69.0 49.6 58.1 61.7 66.3 78.1 62.9 72.4 57.8 61.4 58.0 67.3 88.5 64.6 56.6 60.4 53.1 65.7 75.4 65.7 60.3 62.2 64.3 67.8 61.8 64.9 63.9 61.0 60.4 75.3 62.7 59.9 64.1 63.6 66.1 59.1 72.3 70.3 1997:

56 64 66 67 70 73 85 61 65 68 74 86 92 62 69 77 63 72 78 79 82 1990: 50 58 60 62 64 68 75 59 61 65 78 52 63 69 89 53 66 70 56 57 72 67 1997:

Objective(s) To compute the five-number summary To draw and interpret boxplots

Section 3:5 Answers page 181 #3, 4, 6, 10, 16 #3 a.) skewed right b.) 0, 1, 3, 6, 16 #4 a.) symmetric b.) -1, 2, 5, 8, 11 #6 a.) M =16 b.) Q1 = 22 c.) y has more dispersion d.) Yes, x has an outlier (30) e.) Skewed left #10 a.) 7.2, 9.0, 10.0, 11.2, 16.4

Section 3:5 Answers page 181 #3, 4, 6, 10, 16 #10 b.) c.) skewed right b.) There is not really a difference c.) Keebler has a more consistent number of chips

Objective(s) To compute the five-number summary To draw and interpret boxplots