Priya - Statistics HW 2.2

2.31 (a) Frequency of a data represents the number of times this data value occurred in the data set. Following table shows the frequency distribution of data: x Frequency 0 2 1 5 2 3 4 2 (b) Here, f = 5 represents the frequency 5. It means that data value 1 occurs five times. (c) Sum of the frequency column is 2+5+3+2 = 12 (d) Sum of frequency distribution 12 represents the number of data values in the data set. That is, given data has 12 values. --------------------------------------------------------------------------------------------------------------------- 2.33 (a) For showing the number of goals scored by each player bar graph is appropriate. Histogram is used to show the frequency distribution of a quantitative variable. (b) Following is the bar graph of the data: 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 0 2 4 6 8 10 12 14 16 1 2 2 1 2 8 15 9 1 10 1 6 12 13 1 Goals scored by each player Players Goals (c) Distribution of scoring by the team is the frequency distribution of score. So histogram is more appropriate. (d) Following table shows the frequency distribution of data: x Frequency (f) 1 5 2 3 6 1 8 1 9 1 10 1

12 1 13 1 15 1 Following is the required histogram: 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 0 1 2 3 4 5 6 Distribution of goals by each team Goals Frequency --------------------------------------------------------------------------------------------------------------------- 2.35 (a) The ungrouped frequency distribution for the known team height is, Height in inches Frequency 64 2 65 4 66 4 67 2 68 4 69 1 70 1 Total 18 (b) Using the Excel, the obtained output for the frequency histogram is,

17 1 18 3 19 16 20 10 21 12 22 5 23 1 24 2 (b) Following is the relative frequency distribution of the ages of 50 dancers who responded to a call to audition for a musical comedy: ( ages) Frequency( f ) Relative Frequency Relative Frequency (%) 17 1 1 / 50 = 0.02 2% 18 3 3 / 50 = 0.06 6% 19 16 16 / 50 = 0.32 32% 20 10 10 / 50 = 0.20 20% 21 12 12 / 50 = 0.24 24% 22 5 5 / 50 = 0.10 10% 23 1 1 / 50 = 0.02 2% 24 2 2 / 50 = 0.04 4% Total 50 (c) Following is the relative frequency histogram of the data: 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 T o t a l 0% 5% 10% 15% 20% 25% 30% 35% Relative Frequency Histogram of 50 dancers Dancers Age Frequency (d) Following is the cumulative relative frequency distribution of the test scores:

( ages) Frequency( f ) Cumulative Frequency Cumulative Relative Frequency 17 1 1 1 /50 = 0.02 18 3 4 4 / 50 = 0.08 19 16 20 20 / 50 = 0.4 20 10 30 30 / 50 = 0.60 21 12 42 42 / 50 = 0.84 22 5 47 47 / 50 = 0.94 23 1 48 48 / 50 = 0.96 24 2 50 50 / 50 = 1 Total 50 (e) Following is an ogive of the ages of 50 dancers who responded to a call to audition for a musical comedy: 17 18 19 20 21 22 23 24 25 0 0.2 0.4 0.6 0.8 1 1.2 Distribution of dancers Ages Cumulative Relative Frequency --------------------------------------------------------------------------------------------------------------------- 2.40 (a) Data were collected for the variables time of day when lightning strikes. (b) Each interval represents an hour of the day. For example: 6 am to 7 am, 7 am to 8 am etc. (c) Lightning strikes between 11 am to 12 pm for a maximum number of days. And most of lightning strikes come between the interval 8 am to 5 pm. For some days lightning strikes between 6 am to 7 am. (d) The height of the bars of histogram shows the conclusion. --------------------------------------------------------------------------------------------------------------------- 2.43

--------------------------------------------------------------------------------------------------------------------- 2.45 (a) Following is the grouped frequency distribution by using the class boundaries 12-18, 18-24, …, 48-54. x ( Speeds ) Frequency( f ) 12-18 1 18-24 14 24-30 22 30-36 8 36-42 5 42-48 3 48-54 2 (b) Class width is 18 – 12 = 6 Class width for a class in a frequency distribution is found by subtracting the lower class limit of one class from the lower class limit of the next class. (c) Mid-point of the class 24-30 is 24 + 30 / 2 = 27 Lower boundary of the class is 24 and upper boundary of the class is 30. (d) Following is the frequency histogram of the data:

--------------------------------------------------------------------------------------------------------------------- 2.49 (a) Following is the grouped frequency distribution for the average revenue per kilowatt- hour using classes 4-5, 5-6…. Classes Frequency( f ) 4-5 1 5-6 9 6-7 10 7-8 12 8-9 4 (b) Class width is 5 – 4 = 1 (c) Following are the midpoints associated with the frequency distribution: Classes Mid points 4-5 4 + 5 / 2 = 4.5 5-6 5 + 6 / 2 = 5.5 6-7 6 + 7 / 2 = 6.5 7-8 7 + 8 / 2 = 7.5 8-9 8 + 9 / 2 = 8.5 (d) Following is the relative frequency distribution for the average revenue per kilowatt- hour: Classe s Frequency( f ) Relative Frequency Relative Frequency (%) 4-5 1 1/36 = 0.03 3% 5-6 9 9/36 = 0.25 25%

25 - 35 37 37 + 12 = 49 49/100 = 0.49 35 - 45 26 49 + 26 = 75 75/100 = 0.75 45 - 55 19 75 + 19 = 94 94/100 = 0.94 55 – 65 6 55 + 65 = 100 100/100 = 1 Total 100 (c) The steps to obtain ogive curve are given as follows: 1. Enter the lower bounds of each class and 65 in separate column and cumulative relative frequency in separate column. The table containing the data is given as follows: Annual Salary Cumulative Relative Frequency 15 0 25 0.12 35 0.49 45 0.75 55 0.94 65 1 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 30 40 50 60 70 Y-Values Y-Values (d) The cumulative relative frequency for class 35-45 is 0.75. Therefore, the value that bounds the relative frequency of 0.75 is 45. Converting the bound into thousands, it will be $45,000. (e) The ogive curve shows that the 75% of annual salaries are below $45,000., that is, 75% of annual salaries bounded by $45,000. Both the statements in subpart (d) and (e) represent the same meaning. --------------------------------------------------------------------------------------------------------------------- 2.54 (a) Following is the cumulative relative frequency distribution for the variable “AP scores”:

x ( AP scores) Frequency( f ) Cumulative Frequency Cumulative Relative Frequency 1 9 9 9/42 = 0.21 2 11 20 20/42 = 0.48 3 13 33 33/42 = 0.79 4 6 39 39/42 = 0.93 5 3 42 42/42 = 1 Total 42 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 0.2 0.4 0.6 0.8 1 1.2 Y-Values Y-Values (c) Cumulative relative frequency for the score of 2 is 0.48. (d) Frequency for score of at least 3 will be equal to 100% minus cumulative relative frequency for score of 2. So, 100% - (48%) = 52% (e) Both answers are the same.