Question

1. The following data represent the scores on a test for 36 students in a Sociology class, followed by a stem plot for the data.

65 76 83 81 74 78 41 57 82 50 90 81 63 68 80 85 76 73 79 48 80 75 72 70 75 74 94 87 81 78 64 83 77 72 79 86

4 1 8 5 0 7 6 3 4 5 8 7 0 2 2 3 4 4 5 5 6 6 7 8 8 9 9 8 0 0 1 1 1 2 3 3 5 6 7 9 0 4   

X = ______________________________________________________

ON ALL OF THE FOLLOWING PROBLEMS, SHOW ALL WORK AND LABEL YOUR ANSWERS WITH THE APPROPRIATE UNITS. LABEL ALL AXES ON ANY GRAPHS.

a. Find: median mode(s) range





------------------------------------------------------------------------------------------------------------------------------------------- b. Find and interpret the value for Q1. Show all work.
value of Q1 interpretation







------------------------------------------------------------------------------------------------------------------------------------------- c. Find the values ONLY of Q3 and P85. Show all work.
value of Q3 value of P85

(#1 cont’d)
!!!! DON’T FORGET TO PUT UNITS ON YOUR ANSWERS AND LABEL AXES ON GRAPHS !!!!


d. Give a five-number summary e. Construct a box plot for the test scores. for the test scores.

position value
__________ _____________
__________ _____________
__________ _____________
__________ _____________ _____________________________________________
__________ _____________

----------------------------------------------------------------------------------------------------------------------------------------------- f. Calculate the fences and list the outlier(s) in this set of scores. Construct a modified box plot to clearly show the outlier(s).

calculations lower fence ___________
upper fence ___________
outliers ________________






-------------------------------------------------------------------------------------------------------------------------------------------- g. Construct a modified box plot to clearly show the outlier(s). Label ALL critical positions with their corresponding values.










___________________________________________________________________

2. A random sample of 8 of the tests was chosen and their scores are given below. Calculate the sample mean, the sample variance and the sample standard deviation. Show all work.

78 64 81 75 50 85 79 72

mean








--------------------------------------------------------------------------------------------------------------------------------------------------
variance   

























-------------------------------------------------------------------------------------------------------------------------------------------------- standard deviation

3. The five number summary and the modified box plot for a set of test scores are given below.

five number summary: 33 68 80 86 97

  

modified box plot: 68 80 86 33 38 40 44 _______________________ 97

* * * | | _______________________ ______________________________________________________________________________________ 30 40 50 60 70 80 90 100 test scores   


X = _____________________________________________________________



a. What percent of the students scored at least an 86? ________________


b. What percent of the scores is between 33 and 80? ________________


c. The score of 80 is which of the following? mean median Q1 Q3


d. What percent of the scores is 80 or less? ________________


e. The minimum score for this distribution is ____________ and the maximum is _____________ .


f. List all outliers. ___________________________________


g. What is the value of Q3? ______________


h. Find the fences. Show all work. lower _____________ upper _____________

4. The average height of a 6th grade boy is 58 inches with a standard deviation of 2 inches. The average height of a 9th grade boy is 65 inches with a standard deviation of 3 inches. Both height distributions are bell-shaped.

a. If a 9th grade boy is 60 inches, which of the following is true based on the z-score for his height? Show work.
_____ His height is 1.67 inches below the mean. _____ His height is 1.67 inches above the mean. _____ His height is 1.67 standard deviations below the mean. _____ His height is 1.67 standard deviations above the mean.
-------------------------------------------------------------------------------------------------------------------------------------------------- b. If a 6th grade boy is 55 inches, which of the following is true based on the z-score for his height? Show work.
_____ His height is 1.5 inches below the mean. _____ His height is 1.5 inches above the mean. _____ His height is 1.5 standard deviations below the mean. _____ His height is 1.5 standard deviations above the mean.
-------------------------------------------------------------------------------------------------------------------------------------------------- c. Who is taller for his age: the 55-inch tall 6th grade boy or the 60-inch tall 9th grade boy? _____ The 9th grader is taller because his height is farther from the mean than the 6th grader’s. _____ The 9th grader is taller for his age because 60 inches is greater than 55 inches. _____ The 6th grader is taller because his height is closer to the mean than the 9th grader’s. _____ The 6th grader is taller because his height is not as far below the mean as the 9th grader’s.

-------------------------------------------------------------------------------------------------------------------------------------------------- d. Using the Empirical Rule, approximately ______ percent of 9th grade boys have heights between 59 inches and 71 inches. Show work to support your answer.







-------------------------------------------------------------------------------------------------------------------------------------------------- e. Using the Empirical Rule, approximately 99.7% of 6th grade boys fall between heights of _______________ and ______________. Show work to support your answer

Answer 1

1.a) In order to find the median mode and range, we need to arrange the values in ascending order.

The data is

41,48,50,57,63,64,65,68,70,72,72,73,74,74,75,75,76,76,77,78,78,79,79,80,80,81,81,81,82,83,83,85,86,87,90,94

Median - This is the middle most value in the dataset. Since there are 36 values so there are 2 middle values. The values with index 18 and 19 are the two middle values.

18th indexed number = 76

19th indexed number = 77

Median = (76+77)/2 = 76.5

Mode - The value in the dataset which occurs the most number of times is called the mode of the data. In the above data, 81 occurs thrice and it's the most occurring data value. Hence, the mode is 81.

Range - It is given by the difference between the minimum and the maximum value.

Maximum Value - 94

Minimum Value - 41

Range = 94- 41 = 53

b) Q1 is the first quartile of a dataset. It is given by the 25th percentile. This means that 25% of the values in the dataset will be less than Q1.

In the dataset, the middle value of the first half of the dataset will be called the first quartile. The first half comprises data values indexed from 1 to 18. The 9th and 10th indexed data values will be in the middle of the first half.

9th indexed number = 70

10th indexed number = 72

Q1 = average of the 9th and 10th indexed values = (70+72)/2 = 71

Hence, 25% of the values in the dataset will be less than 71.

c) Q3 is the third quartile of a dataset. It is given by the 75th percentile. This means that 75% of the values in the dataset will be less than Q3.

In the dataset, the middle value of the second half of the dataset will be called the third quartile. The second half comprises data values indexed from 19 to 36. The 27th and 28th indexed data values will be in the middle of the second half.

27th indexed number = 81

28th indexed number = 81

Q3 = average of the 27th and 28th indexed values = (81+81)/2 = 81

The 85th percentile can be calculated by first calculating the index of the value corresponding to the 85th percentile.

Calculation of index i corresponding to the 85th percentile is

Since i is not an integer, hence we would round it up. So i = 31

This means that the number indexed 31 in the dataset will be the value of the 85th percentile.

P85 = 83

d) The 5-number summary for the dataset is

Minimum Value - 41

First Quartile - 71

Median - 76.5

Third Quartile - 81

Maximum Value - 94

Thank You!! Please Upvote!!