Question: Color and clarity of diamonds.Diamonds are categorized according to the “four C’s”: carats, clarity, color, and cut. Each diamond stone that is sold on the open market is provided a certificate by an independent diamond assessor that lists these characteristics. Data for 308 diamonds were extracted from Singapore’s Business Times (Journal of Statistics Education, Vol. 9, No. 1, 2001). Color is classified as D, E, F, G, H, or I, while clarity is classified as IF, VVS1, VVS2, VS1, or VS2. In addition to color and clarity, the independent certification group (GIA, HRD, or IGI), the number of carats, and the asking price were recorded.

a.Use a graphical technique to summarize the color and clarity of the 308 diamond stones. What is the color and clarity that occurs most often? Least often?

b.Use a graphical method to describe the carat distribution of all 308 diamonds.

c.Use a graphical method to describe the carat distribution of diamonds certified by the GIA group.

d.Repeat part cfor the HRD and IGI certification groups.

e.Compare the three-carat distributions, parts cand d.Is there one particular certification group that appears to be assessing diamonds with higher carats than the others?

f.Find and interpret the mean of the data set.

g.Find and interpret the median of the data set.

h.Find and interpret the mode of the data set.

i.Which measures of central tendency best describe the 308-carat values? Explain.

j.Use the mean and standard deviation to form an interval that will contain at least 75% of the carat values in the data set.

k.Construct a scatter plot for the data, with a number of carats on the horizontal axis and price on the vertical axis. What type of trend do you detect?

Step by step solution

01

Given information

The data was collected from Singapore’s business time. In addition to color and clarity, the independent certification groups, the number of carats, and the asking price were recorded.

02

(a) Colour and clarity most often and least often

It is most often when the price is at the same level and least when the price decreases. The most often occurring color is F. D is the least frequent color. The most common type of clarity is VS1. VVS1 is the least common clarity.

03

(b) Graphical method to describe the carat distribution

In this case, carat distribution follows symmetric distribution, which means it follows a normal distribution.

04

(c) Graphical method to describe the carat distribution of diamonds certified by the GIA group

The carat distribution of diamonds follows the normal distribution, certified by the GIA group.

05

(d) Find the distribution of diamonds certified by HRD and IGI certification groups.

Similarly, following the previous two steps, they also follow a normal distribution.

06

(e) Compare the three-carat distributions and define which diamond has more carats than others

To compare three-carat distributions, they follow the same distributions. HRD has more carats than GIA and IGI has more carats than HRD. In these three, IGI has the highest carat than GIA and HRD.The GIA appears to be appraising larger carat diamonds than the other organizations.

07

(f) Find intercept and mean of the distribution

Intercept is color and clarity. The mean of the data set is:

$\overline{x}=0.6309091$

08

(g) Find intercept and median of the distribution

The median of the distribution is 0.62

09

(h) Find intercept and mode of the distribution

The mode of the distribution is 1. Because the distribution is symmetric.

10

(i) Type of central tendency and explain

For carat values, the measure of central tendency used here is mean. To find the average value. Although our information is slightly shifted to the right, the mean is bigger than the median. Since our data set contains outliers, the median is the best indicator of central tendency for the 308-carat values.

11

(j) Find the mean and standard deviation at least of the carat values in the dataset

For 75% of the dataset the value is given by

$$\begin{gathered} \overline{x}=\text{0.6309091} \\ S=\sqrt{0.0768304} \\ =0.2771829721 \\ \left(0.6309091-\text{Z}\left(0.2771829721\right.\right),\left(0.6309091+\text{Z}\left(0.2771829721\right.\right) \\ (0.077,1.185) \end{gathered}$$

12

(k) Draw the scatterplot and select the type of data

The data is correlated type data.

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