The following histogram shows the scores on the first exam for a stati
The following histogram presents the scores obtained by students in their first statistics exam. To analyze this data comprehensively, several questions need to be addressed, encompassing total students, class intervals, midpoints, and specific frequency counts.
1. How many students took the exam?
To determine the total number of students who took the exam, we need to sum the frequencies across all classes displayed in the histogram. The histogram provides visual counts or frequencies for each score range, which can be added together to find the total student count. For instance, if the histogram shows the following frequencies for each class: 8 students scored within 60-69, 12 within 70-79, 15 within 80-89, and 10 within 90-100, summing these gives the total number of students: 8 + 12 + 15 + 10 = 45 students. Since specific values are not provided here, the precise total should be computed from the histogram data, but the method involves summing all class frequencies.
2. What is the class interval?
The class interval refers to the range of scores within each class in the histogram. Typically, this is the difference between the upper and lower bounds of a class. For example, if the classes are 60-69, 70-79, etc., then the class interval is 10 points. In this data, the class interval is likely 10, as suggested by standard grouping in exam score distributions, unless the histogram specifies otherwise.
3. What is the class midpoint for the first class?
The class midpoint is calculated by averaging the lower and upper bounds of the class. For a class 60-69, the midpoint is (60 + 69)/2 = 64.5. This value is useful for constructing frequency polygons or calculating measures such as the mean.
4. How many students earned a score of less than 70?
To find the number of students who scored below 70, sum the frequencies of all classes with upper bounds less than 70. For example, if the data shows 8 students in 60-69 and 5 students in 50-59, then total students scoring less than 70 would be 8 + 5 = 13 students. Exact figures can be derived from the histogram's class frequencies.
5. For a data set of 83 observations, how many classes are recommended?
Number of classes can be estimated using Sturges’ Rule:
k = 1 + 3.322 log
10 (n) . For n=83,
k ≈ 1 + 3.322 * log
10
(83) ≈ 1 + 3.322 * 1.919 ≈ 1 + 6.37 ≈ 7.37.
Thus, approximately 7 or 8 classes are recommended for a frequency distribution.
This dataset depicts the number of packages shipped daily over 100 days. We analyze it by identifying the type of chart, total frequencies, class intervals, class frequencies, and specific counts, like days with shipments of 20 or more.
a. What is this chart called?
The chart illustrating the number of packages shipped per day over 100 days, with the frequencies per class, is called a histogram. A histogram displays frequency distributions of continuous data effectively, showing the spread and shape.
b. What is the total number of frequencies?
The total number of frequencies corresponds to the total days observed, which is 100 days, as each day’s shipments fall into one class interval.
c. What is the class interval?
Class interval refers to the range each class covers in the histogram. For shipment data, if classes are, for example, 0-4, 5-9, 10-14, etc., the class interval is 5 packages.
d. What is the class frequency for the 0 up to 5 class?
The class frequency indicates how many days had shipments between 0-4 packages. Suppose the histogram shows that for this class, 15 days fell into this range; thus, the class frequency is 15.
e. What is the relative frequency of the 0 up to 5 class?
The relative frequency is calculated as the class frequency divided by the total number of observations: Relative frequency = 15/100 = 0.15. Rounded to two decimal places, it remains 0.15.
f. What is the midpoint of the 20 up to 25 class?
The midpoint is (20 + 25) / 2 = 22.5. This value is crucial for constructing frequency polygons.
g. On how many days were there 20 or more packages shipped?
If classes 20-24, 25-29, etc., include days with 20 or more packages, summing their frequencies gives the total days with 20+ shipments. For example, if these classes encompass 40 days, then 40 days had 20 or more packages shipped.
The data ranges from a minimum of 42 to a maximum of 129, with 53 total observations. To effectively represent this data, determining the number of classes and the lower limits is essential.
a. How many classes would you suggest?
Using Sturges' rule: k = 1 + 3.322 log
10
(53) ≈ 1 + 3.322 * 1.724 ≈ 1 + 5.73 ≈ 6.73. Round to 7 classes to suitably organize the data.
b. What would you suggest as the lower limit of the first class?
Given the minimum value is 42, the lower limit of the first class should be 40 to 41, ensuring the data is adequately covered.
In a pie chart representing market share with a Pepsi-Cola slice having a 90-degree central angle, the percentage market share can be calculated as:
Market share % = (Central angle / 360°) * 100 = (90 / 360) * 100 = 25%.
Hence, Pepsi-Cola holds a 25% market share.
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