Answered

The following table shows the number of hours some high school students in 2 cities spend searching the Internet each week:

\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
City A & 35 & 36 & 35 & 34 & 35 & 38 & 36 & 36 & 38 \\
\hline
City B & 22 & 20 & 50 & 25 & 24 & 65 & 25 & 26 & 24 \\
\hline
\end{tabular}

Part A: Create a five-number summary and calculate the interquartile range for the two sets of data. (6 points)

Part B: What is the best measure of center? Justify your answer. (4 points)

Total: (10 points)



Answer :

GinnyAnswer
Let's address the problem step-by-step:

### Part A: Five-number summary and Interquartile Range (IQR)

#### City A:
The data for City A are: 35, 36, 35, 34, 35, 38, 36, 36, 38

1. Minimum (Min): The smallest number in the data set.
- Min for City A: 34

2. First Quartile (Q1): The 25th percentile of the data set.
- Q1 for City A: 35.0

3. Median: The middle number in the sorted data set.
- Median for City A: 36.0

4. Third Quartile (Q3): The 75th percentile of the data set.
- Q3 for City A: 36.0

5. Maximum (Max): The largest number in the data set.
- Max for City A: 38

Therefore, the five-number summary for City A is:
- Min: 34
- Q1: 35.0
- Median: 36.0
- Q3: 36.0
- Max: 38

6. Interquartile Range (IQR): The difference between the third quartile (Q3) and the first quartile (Q1).
- IQR for City A: [tex]\(36.0 - 35.0 = 1.0\)[/tex]

#### City B:
The data for City B are: 22, 20, 50, 25, 24, 65, 25, 26, 24

1. Minimum (Min): The smallest number in the data set.
- Min for City B: 20

2. First Quartile (Q1): The 25th percentile of the data set.
- Q1 for City B: 24.0

3. Median: The middle number in the sorted data set.
- Median for City B: 25.0

4. Third Quartile (Q3): The 75th percentile of the data set.
- Q3 for City B: 26.0

5. Maximum (Max): The largest number in the data set.
- Max for City B: 65

Therefore, the five-number summary for City B is:
- Min: 20
- Q1: 24.0
- Median: 25.0
- Q3: 26.0
- Max: 65

6. Interquartile Range (IQR): The difference between the third quartile (Q3) and the first quartile (Q1).
- IQR for City B: [tex]\(26.0 - 24.0 = 2.0\)[/tex]

### Part B: Best Measure of Center

To determine the best measure of center, we should consider the presence of outliers and the distribution of data.

#### City A:
- The five-number summary for City A: (34, 35.0, 36.0, 36.0, 38)
- The IQR for City A is relatively small (1.0), indicating a more consistent data set with no significant outliers.

For City A, the data set is fairly consistent without outliers. Therefore, both the mean and the median could be reasonable measures of center. However, we commonly take the median as a strong measure of center in such datasets.

Best measure of center for City A: Median [tex]\(36.0\)[/tex]

#### City B:
- The five-number summary for City B: (20, 24.0, 25.0, 26.0, 65)
- The presence of a large maximum value (65) indicates there is an outlier in the data set. Such outliers can skew the mean, making it a less reliable measure of center.

For City B, the presence of the outlier (65) suggests that the median is a more robust and reliable measure of center than the mean.

Best measure of center for City B: Median [tex]\(25.0\)[/tex]

### Summary:

- City A:
- Five-number summary: (34, 35.0, 36.0, 36.0, 38)
- Interquartile Range (IQR): 1.0
- Best measure of center: Median (36.0)

- City B:
- Five-number summary: (20, 24.0, 25.0, 26.0, 65)
- Interquartile Range (IQR): 2.0
- Best measure of center: Median (25.0)

This concludes the detailed step-by-step solution for the given question.