Q 38. - The Practice of Statistics for AP

Question

Olympic gold! The following table displays the total number of gold medals won by a sample of countries in the $2008$ Summer Olympic Games in China.

(a) Make a dot plot to display these data. Describe the overall pattern of the distribution and any deviations from that pattern.

(b) Overall, $204$ countries participated in the $2008$ Summer Olympics, of which $55$ won at least one gold medal. Do you believe that the sample of countries listed in the table is representative of this larger population? Why or why not? 

Step-by-Step Solution

Answer

Part (a) 

• Shape: Because the highest dots are to the left of the dot-plot, the data is right-skewed.
• Centre:  Because the largest dots in the dot-plot are at $0$, it's about $0$ gold medals in the center.
• Spread: The range is $0$ hours to $51$ gold medals.
• Outliers: The data contains three outliers: China's, the United Kingdom's, and the United States' values.

Part (b) The given sample does not represent the $2008$ Summer Olympics.

Part (a) Step 1: Given information

The table is 

Country	Gold medal	Country	Gold medals
Sri Lanka	0	Thailand	2
Vietnam	0	Kuwait	0
Great Britain	19	Bahamas	0
Norway	3	Kenya	5
Romania	4	Trinidad and Tobago	0
Switzerland	2	Greece	0
Armenia	2	Mozambique	0
Netherlands	7	Kazakhstan	2
India	0	Denmark	2
Georgia	3	Latvia	1
Kyrgyzstan	0	Czech Republic	3
Costa Rica	0	Hungary	3
Brazil	3	Sweden	0
Uzbekistan	1	Uruguay	0
China	51	United states	36

Part (a) Step 2: Concept

Shape: The peak of the dot plot is at  This means that during that season, the team's most common number of goals scored in games (known as the mode) was  The United States women's soccer team scored between one and five goals in the majority of its games. The distribution, however, has a large tail to the right. 

Centre: can be described by selecting a value that divides the observations in half, with the bigger half taking greater values and the smaller half taking lower values. 

The spread of distribution indicates the amount of variability in the data. Giving the smallest and greatest values is one approach to describe the variability. 

An outlier is a value that is notably different from other values in a random sampling of the population.

Part (a) Step 3: Calculation

We have the following information on the total number of gold medals earned by a randomly selected sample of countries: 

Country	Gold medal
	0
	51
	0
	19
	3
	4
	2
	0
	7
	0
	3
	0
	0
	3
	1
	2
	0
	0
	5
	0
	0
	0
	2
	2
	1
	3
	3
	0
	0
	36

A dot plot is a simple graph in which each data value is represented by a dot above its location on a number line. The following is a dot plot of the given data: 

Country	Gold medal	Country	Gold medal
Sri Lanka	0	Thailand	2
Vietnam	0	Kuwait	0
Great Britain	19	Bahamas	0
Norway	3	Kenya	5
Romania	4	Trinidad and Tobago	0
Switzerland	2	Greece	0
Armenia	2	Mozambique	0
Netherlands	7	Kazakhstan	2
India	0	Denmark	2
Georgia	3	Latvia	1
Kyrgyzstan	0	Czech Republic	3
Costa Rica	0	Hungary	3
Brazil	3	Sweden	0
Uzbekistan	1	Uruguay	0
China	51	United states	36

The shape, center, spread, and outliers of distribution can all be used to define its general pattern.

Shape: The Dot plot's peak is $0$; this signifies that none of the countries in the maximum sample won gold. As a result, $0$ is the mode. Between $0$and $3$ gold medals have been earned by the majority of countries. The distribution, on the other hand, has a long slope to the right. 

The center can be described by selecting a value that splits the data in half. As a result, roughly half of the people choose greater values while the other half choose smaller values. This is known as the Median of the Distribution, which in this case is $3$. For this data, we can also determine the mean number of hours, which is $4.9$

The spread of the distribution indicates the degree of variability in the data. Giving the smallest and greatest values is one approach to describe the variability. The time span of the data ranges from $0$ to $51$ hours. The range is determined by subtracting the lowest value from the biggest value.

$51 Gold medals- 0 Gold medals= 51 Gold medals$

Outliers are values that deviate significantly from the overall pattern and stand out from the rest of the distribution. The outliers are ages $19, 36,$ and $51$ Outliers must result in differences in the mean, median, and mode.

The mean $(4.9)$ is greater than the Median $\left(3\right)$,  which is greater than Mode $\left(0\right)$

Shape: Because the highest dots are to the left of the dot-plot, the data is right-skewed.

Centre:  Because the largest dots in the dot-plot are at $0$, it's about $0$ gold medals in the center.

Spread: The range is $0$ hours to $51$ gold medals.

Outliers: The data contains three outliers: China's, the United Kingdom's, and the United States' values.

Part (b) Step 1: Calculation

We have the following information on the total number of gold medals earned by a randomly selected sample of countries: 

Country	Gold medal
	0
	51
	0
	19
	3
	4
	2
	0
	7
	0
	3
	0
	0
	3
	1
	2
	0
	0
	5
	0
	0
	0
	2
	2
	1
	3
	3
	0
	0
	36

We've already mentioned that $55$ of the $204$ countries that competed in the $2008$ Summer Olympics earned at least one gold medal. According to standard criteria, the percentage of countries that have won at least one gold medal is expressed as

We can see that $17$ countries have earned at least one gold medal in the given sample of $30$ countries. According to standard criteria, the percentage of countries that have won at least one gold medal is expressed as 

$55/204=26.96\%$

As a result, the sample provided does not accurately reflect the $2008$ Summer Olympics.