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College students and surfers Rex Robinson and Sandy Hudson collected data on the self-reported numbers of days surfed in a month for 30 longboard surfers and 30 shortboard surfers. Longboard: \(4,9,8,4,8,8,7,9,6,7,10,12,12,10,14,12,15,13\), \(10,11,19,19,14,11,16,19,20,22,20,22\) Shortboard: \(6,4,4,6,8,8,7,9,4,7,8,5,9,8,4,15,12,10,11,12\), \(12,11,14,10,11,13,15,10,20,20\) a. Compare the typical number of days surfing for these two groups by placing the correct numbers in the blanks in the following sentence: The median for the longboards was for the shortboards was \(\quad\) days, showing that those with boards typically surfed more days in this month. b. Compare the interquartile ranges by placing the correct numbers in the blanks in the following sentence: The interquartile range for the longboards was shortboards was \(\quad\) days, showing more variation in the days surfed this month for the boards.

Step by step solution

01

- Organize the Data

The first thing to do is to organize the data for both longboard and shortboard surfers in ascending order. This will allow us to identify the median and the interquartile range (IQR) more easily.

02

- Calculate the Medians

To calculate the median, find the middle number in the ordered list of values. If there are two middle numbers, calculate their average. Do this both for longboard and shortboard surfers.

03

- Calculate the Interquartile Ranges

The interquartile range (IQR) is calculated by subtracting Q1, the 25th percentile, from Q3, the 75th percentile. To find these percentiles, divide your ordered data list into four equal parts. Q1 will be the median of the first half of your data, and Q3 will be the median of the second half. Calculate the IQRs for both the longboard and shortboard surfers.

04

- Compare the Medians and IQRs

Now, complete the sentence prompts with the calculated medians and IQRs. Interpret these values to see which group typically surfs more days in a month and which group has more variation in the number of days surfed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Data Collection

Data collection is a crucial preliminary step in statistical analysis. It involves systematically gathering information relevant to a particular field of study or interest. For our surfing enthusiasts, Rex Robinson and Sandy Hudson, the method of data collection was a self-reported survey where they asked longboard and shortboard surfers to state the number of days they surfed in a month.

Accuracy in data collection is paramount as it affects the reliability of the results. In educational settings, teaching students effective data collection methods is vital. These methods include surveys, experiments, observational studies, and databases. Ensuring that data collected are representative of the population and free from biases sets the stage for meaningful analysis.

Median Calculation

The median is a measure of central tendency that identifies the middle value in a data set when arranged in ascending or descending order. When the dataset has an odd number of observations, the median is the middle number. However, if there is an even number of observations, as in the case of our 30 surfers in each group, the median is calculated by averaging the two middle numbers.

To help students grasp this concept better, it’s useful to explain that the median divides the data into two halves – 50% of the scores lie below the median and 50% above. This is particularly helpful when dealing with skewed data or when there are outliers, as the median provides a better central measure than the mean.

Interquartile Range (IQR)

The interquartile range (IQR) is a measure of variability that indicates the spread of the middle 50% of data. It is the difference between the third quartile (Q3) and the first quartile (Q1). Quartiles divide the data set into four equal parts.

Finding Q1 and Q3

To find Q1 (the 25th percentile), divide the ordered dataset into two halves and identify the median of the lower half. For Q3 (the 75th percentile), do the same for the upper half. The IQR is then \( Q3 - Q1 \). The greater the IQR, the more spread out the middle 50% of the data is, which indicates more variability within the dataset.

Percentiles

Percentiles are values that divide a data set into 100 equal parts, and they are useful in comparing individual scores to the broader dataset. For example, if a surfer’s days surfed is at the 80th percentile, this means they surfed more days than 80% of the other surfers.

To calculate percentiles, a data set is ordered from least to greatest. The position of a percentile is determined by the formula: \(P = \frac{N + 1}{100} * K\) where \(P\) is the percentile's position, \(N\) is the total number of observations, and \(K\) is the percentile number. Teaching percentiles is key for students to understand how individual observations relate to the overall distribution.

Data Interpretation

Data interpretation refers to the process of analyzing and making sense of collected data. It involves extracting meaningful insights and recognizing patterns or trends. In the context of the exercise, interpreting the median and IQR involves comparing these measures for the two groups of surfers to discern their surfing habits.

Effective interpretation requires critical thinking to understand what the numbers actually signify about the real world. For students, developing this skill areas involves practice in looking beyond the numbers to the context of the data. For instance, a higher median number of surf days for one group may imply a greater enthusiasm or availability for surfing, while a larger IQR could suggest that this group's surf habits are more diverse.