87. College Football A stem-and-leaf plot for the numbers of touchdowns allowed by the 127 NCAA Division I Football Bowl Subdivision teams in the 2020-2021 season is shown. Find the probability that a team chosen at random allowed (a) at least 51 touchdowns. Are any of these events unusual? Explain. (Source: National Collegiate Athletic Association)

Verified step by step guidance

1. Step 1: Understand the stem-and-leaf plot. Each row represents the tens digit (stem), and each number in the row represents the units digit (leaf). For example, in the row labeled '5', the numbers 0, 2, 3, 6, 6, 8, and 8 represent 50, 52, 53, 56, 56, 58, and 58 touchdowns respectively.
2. Step 2: Identify the teams that allowed at least 51 touchdowns. This includes all values in the stem '5' and any values greater than 50 in the stem '4'. Count the total number of these values.
3. Step 3: Calculate the total number of teams. Count all the leaves across all rows to determine the total number of teams (127 as stated in the problem).
4. Step 4: Compute the probability. Divide the number of teams that allowed at least 51 touchdowns by the total number of teams. Use the formula: \( P(X \geq 51) = \frac{\text{Number of teams with touchdowns} \geq 51}{\text{Total number of teams}} \).
5. Step 5: Determine if the event is unusual. An event is considered unusual if its probability is less than 0.05. Compare the calculated probability to 0.05 and explain whether the event is unusual or not.

Key Concepts

Stem-and-Leaf Plot

A stem-and-leaf plot is a graphical representation used to display quantitative data in a way that retains the original data values while showing their distribution. Each number is split into a 'stem' (the leading digit or digits) and a 'leaf' (the trailing digit). This format allows for easy visualization of the data's shape and helps identify the frequency of values within specific ranges.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it involves calculating the proportion of teams that allowed at least 51 touchdowns out of the total number of teams. Understanding how to compute probabilities is essential for making inferences about data and assessing the likelihood of various outcomes.

Unusual Events

An event is considered unusual if its probability is significantly low, often defined as less than 5%. In the context of this question, determining whether the event of a team allowing at least 51 touchdowns is unusual involves calculating its probability and comparing it to this threshold. This concept helps in understanding the rarity of certain outcomes within a dataset.