Question

Here is a set of data of college GPAs from a sample of students who completed an AP course in statistics in their senior year at a fictional high school: (9 points) 3.8, 3.8, 3.9, 3.9, 3.5, 3.5, 3.0, 3.5, 3.8, 4.0, 3.5, 2.5, 2.4, 1.9, 4.0, 3.8, 3.7, 3.5, 3.9, 3.2 A. Construct a histogram of the GPA data using a class width of 0.3, and a minimum x-value of 1.9, and be sure to label both axes. B. Construct a modified box-and-whisker plot using the GPA data and report the values of the five-number summary, labeling the axes, and labeling Q1, Q3, the median and the mean on the boxes. C. Describe the shape of the distribution of GPAs in terms of symmetry (is it skewed?) and outliers. Use evidence from the two types of plots to support your answer

Answer 1

A. In this representation, the frequency of GPAs falling within each range is indicated, and the ranges are defined by the class width of 0.3.

B. Modified box-and-whisker plot:

• Five-number summary:
• Minimum: 1.9
• Q1: 3.1
• Median: 3.7
• Q3: 3.8
• Maximum: 4.0

C. Description of the shape of the distribution:

• Symmetry: The distribution appears slightly skewed to the left, as the left tail is longer than the right in both the histogram and the box-and-whisker plot.
• Outliers: There are potential outliers on the lower end, as evidenced by the values below the left whisker in the box-and-whisker plot. These outliers could be contributing to the left skewness.

1. Class Width and Minimum X-value:

The class width is given as 0.3, and the minimum x-value is 1.9. Therefore, your first class interval will be [1.9, 2.2). Subsequent intervals will increase by 0.3 each, creating classes like [2.2, 2.5), [2.5, 2.8), and so on.

2. Counting Frequencies:

Count the number of data points that fall within each class interval. For example, there are two data points in the [3.5, 3.8) interval, four points in the [3.8, 4.1) interval, and so on.

3. Drawing the Histogram:

On the x-axis, mark the interval boundaries (e.g., 1.9, 2.2, 2.5, etc.). On the y-axis, scale the height of the bars to represent the frequency of data points in each interval. For example, if there are three data points in the [3.2, 3.5) interval, the corresponding bar would be three times taller than the bar for the interval with only one data point.

4. Labeling Axes:

Label the x-axis as "GPA" and the y-axis as "Frequency."

B. Modified Box-and-Whisker Plot

1. Five-Number Summary:

To create the box-and-whisker plot, you first need to calculate the five-number summary:

Minimum: 1.9

Q1 (First Quartile): 3.25

Median: 3.65

Q3 (Third Quartile): 3.9

Maximum: 4.0

2. Drawing the Box:

Draw a box representing the interquartile range (IQR) from Q1 to Q3.

Draw a horizontal line inside the box for the median.

3. Drawing the Whiskers:

Extend the whiskers from the ends of the box to the minimum and maximum values, but only if they are within 1.5 times the IQR from the Q1 and Q3, respectively.

If any values are further than that, they are considered outliers and should be plotted individually beyond the whiskers.

4. Labeling Axes and Values:

Label the x-axis as "GPA."

Label the box with Q1, Q3, and the median inside.

If there are outliers, mark them as individual points beyond the whiskers.

C. Distribution Shape

Based on the histogram and box-and-whisker plot, you can analyze the shape of the GPA distribution:

Symmetry:

The histogram may show a slight right skew, with more data points towards the lower end of the GPA range. The box-and-whisker plot might also show a slightly longer whisker towards the lower end, further suggesting a right skew.

Outliers:

There might be one or two outliers at the lower end of the distribution, as indicated by data points potentially falling outside the whiskers in the box-and-whisker plot.

Therefore, the distribution of GPAs is slightly right-skewed with the possibility of one or two outliers at the lower end.