Question

Each murder committed in Utah during the period \(1978-1990\) was categorized by day of the week, resulting in the following frequencies: Sunday, 109 ; Monday, 73 ; Tuesday, 97 ; Wednesday, 95 ; Thursday, 83 ; Friday, 107 ; Saturday, 100 . a. Construct the corresponding frequency distribution. b. What proportion of these murders was committed on a weekend day - that is, Friday, Saturday, or Sunday? c. Do these data suggest that a murder is more likely to be committed on some days than on other days? Explain your reasoning.

Short answer

a. Frequency distribution: Sunday, 109 ; Monday, 73 ; Tuesday, 97 ; Wednesday, 95 ; Thursday, 83 ; Friday, 107 ; Saturday, 100. b. The proportion of murders that were committed on a weekend day is 47.6%. c. The per day numbers of murders do not strongly suggest that murders are more likely to be committed on certain days over others.

Step-by-step solution

Construct Frequency Distribution

Construct a frequency distribution by listing all seven days of the week and the corresponding frequencies: Sunday, 109 ; Monday, 73 ; Tuesday, 97 ; Wednesday, 95 ; Thursday, 83 ; Friday, 107 ; Saturday, 100. This distribution illustrates the total number of murders for each day of the week during the 1978-1990 period.

Calculate Weekend Proportion

Now, to figure out the proportion of murders that occurred on weekend days (i.e., Friday, Saturday, or Sunday), the first step is to add up the number of murders that occurred on these days. The weekend total is thus 109 (Sunday) + 107 (Friday) + 100 (Saturday) = 316. The next step is to compute the total number of murders for the entire week, which is 109 + 73 + 97 + 95 + 83 + 107 + 100 = 664. Thus, the proportion of murders that occurred on weekend days is \(316/664 = 0.476\), or 47.6% when expressed as a percentage.

Assess Daily Likelihood

To gauge whether a murder is more likely on certain days, the data can be examined again. While the highest number of murders occurred on Sundays (109) and the lowest on Mondays (73), these numbers by themselves are not massively different. Therefore, on face value, the data does not suggest a strongly greater likelihood of a murder happening on a particular day of the week over another.

Key concepts

Descriptive Statistics

Descriptive statistics involve summarizing and organizing data in a way that is useful and informative. The fundamental aspect of descriptive statistics is to present data in a manageable form. When dealing with a large data set, as in the exercise with murders in Utah, simplifying the information through a frequency distribution is a typical method.

Frequency distributions list the different values (in this case, days of the week) along with their corresponding frequencies (the number of murders on each day). This type of table makes it possible to see at a glance which days had more occurrences of a particular event. When it comes to interpreting this information, descriptions such as measures of central tendency (mean, median, mode) and measures of variability (range, standard deviation) could also be considered; however, in the given exercise, the focus is on the frequency of events over the period.

Using a frequency distribution, like in Step 1 of the solution, allows students not only to organize their data but also to form a visual picture, sometimes leading to bar graphs or histograms that represent the data visually. This is an essential skill in both academic and real-world settings where the ability to analyze trends or patterns quickly can lead to more informed decisions. Hence, introducing students to the basics of constructing and analyzing frequency distributions is a crucial component of their education in statistics.

Data Analysis

Data analysis is the process of systematically applying statistical and logical techniques to describe, illustrate, condense and evaluate data. In the context of the exercise on Utah's murder data, data analysis begins after creating the frequency distribution and involves interpreting the data to provide answers to specific questions.

Step 3 of the problem-solving process requires a simple form of data analysis, where the student needs to compare the frequencies to determine if there is a pattern that suggests murders are more likely to occur on certain days. This comparison involves looking at the distribution of values and determining whether any are significantly higher or lower than the others. If certain days had frequencies that were particularly high or low in comparison to other days, this could imply patterns of occurrence.

Importance of Context in Analysis

However, what's especially important in data analysis is context. While the frequency can give some indication, further questions arise: Are there external factors influencing these numbers? Could certain events on weekends cause a fluctuation in crimes? Without considering context, it’s possible to make inaccurate assumptions about the data. Thus, even with a relatively straightforward frequency analysis, careful consideration and additional research are often warranted to draw valid conclusions.

Proportion Calculation

Proportion calculation allows us to understand the relative size of one part in comparison to the whole. It is a fundamental concept in statistics, often reported as a percentage, making it easy to comprehend and communicate. The calculation involves two steps: firstly, determining the numerator (the part) and secondly, the denominator (the whole), then dividing the former by the latter.

As illustrated in Step 2 of the solution provided, calculating the proportion of murders on weekend days involves adding the frequencies of murders for Friday, Saturday, and Sunday, then dividing this total by the frequency of murders for all days of the week. The result is a number between 0 and 1, which could be multiplied by 100 to express it as a percentage.

Kinetic Approach to Learning Proportions

For students who find the concept of proportions challenging, employing a kinetic approach to learning could prove beneficial. Engaging with physical representations—like using different colored blocks to represent days of the week and stacking them to visualize the parts versus the whole—might help solidify their understanding of proportions. Such hands-on activities can make abstract concepts more concrete. Always remember, when explaining proportions, clear visual examples and comparisons to real-life scenarios can help students comprehend and remember how to perform these calculations.