Question

Heal the Bay is an environmental organization that releases an annual beach report card based on water quality (Heal the Bay Beach Report Card, May 2009). The 2009 ratings for 14 beaches in San Francisco County during wet weather were: $$ \mathrm{A}+\mathrm{C} \mathrm{B} \mathrm{A} \quad \mathrm{A}+\mathrm{A}+\mathrm{A} \mathrm{A}+\mathrm{B} \mathrm{D} \mathrm{C} \mathrm{D} \mathrm{F} \mathrm{F} $$ a. Summarize the wet weather ratings by constructing a relative frequency distribution and a bar chart. b. The dry weather ratings for these same beaches were: \(\mathrm{A} \mathrm{B} \mathrm{B} \quad \mathrm{A}+\mathrm{A} \mathrm{F} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{A} \quad \mathrm{B} \quad \mathrm{A}\) Construct a bar chart for the dry weather ratings. c. Do the bar charts from Parts (a) and (b) support the statement that beach water quality tends to be better in dry weather conditions? Explain.

Short answer

The answer to whether the beach water quality is better during dry weather conditions would require a comparative analysis of the two bar charts. It would depend on whether the ratings during dry weather are systematically better than those during wet weather.

Step-by-step solution

Wet Weather Ratings Table and Bar Chart

Count the frequency of each rating (A+, A, B, C, D, F) during wet weather, and calculate the relative frequency by dividing each frequency by the total number of ratings. Create a bar chart based on these frequencies.

Dry Weather Ratings Bar Chart

Next, count the frequency of each rating (A+, A, B, C, D, F) during dry weather and create a bar chart based on these frequencies.

Comparative Analysis

Compare the bar charts of wet weather and dry weather. If the scores in the dry weather bar chart are systematically better than the wet weather chart, this would support the statement that beach water quality tends to be better in dry weather conditions.

Key concepts

Relative Frequency Distribution

Relative frequency distribution is a fundamental aspect of statistical data analysis, which involves counting how often each value appears and then expressing these counts as a proportion of the total number of observations. This method allows us to understand the distribution of data points within a dataset, relative to one another.

The process begins by listing all possible categories or ratings that the data can fall into. Once we have this listing, we tally up how many times each category occurs. To find the relative frequency, we divide the frequency of each category by the total number of observations. In mathematical terms, if the frequency of a certain category is represented by the symbol 'f' and the total number of observations is 'N', the relative frequency ('rf') is calculated as:
\[ r_{f} = \frac{f}{N} \]
For example, if we have an environmental study which rates the quality of beach water and 'A+' was received 3 times out of a total of 14 observations, the relative frequency of 'A+' would be \( r_{f}(A+) = \frac{3}{14} \).

Understanding relative frequency distribution is essential, as it provides insights into the prevalence of certain outcomes or characteristics within the data set being analyzed. Furthermore, improving the clarity of this concept to students would involve reinforcing the notion that relative frequency tells us 'how often' something happens compared to the whole, which is especially useful in making comparisons between different datasets or conditions.

Bar Chart Construction

A bar chart is a graphical representation of data that uses bars to show the frequencies or the relative frequencies of different categories. Each bar's length represents the quantity associated with that category, making comparisons among categories straightforward. The construction of a bar chart for our beach water quality example follows these steps:

First, we need to establish the categories or ratings along the horizontal axis (x-axis), which in this case are the different grades (A+, A, B, C, D, F). Each category will have its corresponding bar.
Next, the vertical axis (y-axis) is designated for the frequencies or relative frequencies of each category. It's crucial to use the same scale for all bars for the sake of accurate comparison.
Then, we draw a bar for each rating whose height is proportional to the count or relative frequency of that particular category. These bars are typically spaced equally apart to enhance the chart's readability.

For students to improve their grasp on bar chart construction, it is important to stress the aspect of proportional representation and consistent scaling. They must ensure each bar accurately represents the data it stands for. Color coding or labeling can also be utilized to enhance the understanding and aesthetic of the chart.

Comparative Analysis

Comparative analysis is a crucial part of statistical data analysis, permitting an invaluable insight into how different datasets relate to each other. For instilling a comprehensive understanding, the exercise we're referencing involves comparing water quality during wet and dry weather conditions, utilizing bar charts.

With the bar charts for both conditions prepared, examine how the frequency distributions differ. In a comparative analysis, we look for patterns or notable distinctions like which weather condition resulted in higher quality ratings more frequently.

To conduct a comparative analysis effectively, focus on:

• The direction of the differences: Whether one condition consistently shows higher ratings than the other.
• The magnitude of the differences: How significant the discrepancies between the two conditions are.
• The variety within each distribution: Check if one condition results in a wider or narrower spread of ratings.

When extrapolating findings, it's crucial to consider external factors that might influence the results and to avoid jumping to conclusions based on a simplified view of the data. Enhancing student understanding of comparative analysis requires the promotion of critical thinking, encouraging them to question what might cause the observed differences, thereby deepening their analytical skills.