Question

In a certain population of the freshwater sculpin, Cottus rotheus, the distribution of the number of tail vertehrae is as shown in the table. $$ \begin{array}{|cr|} \hline \text { No. of vertebrae } & \text { Percent of fish } \\ \hline 20 & 3 \\ 21 & 51 \\ 22 & 40 \\ 23 & 6 \\ \hline \text { Total } & 100 \\ \hline \end{array} $$ Find the probability that the number of tail vertebrae in a fish randomly chosen from the population (a) equals 21 . (b) is less than or equal to 22 (c) is greater than 21 . (d) is no more than 21 .

Short answer

(a) 0.51 (b) 0.94 (c) 0.46 (d) 0.54

Step-by-step solution

Part A: Finding the Probability for 21 Vertebrae

To find the probability that a randomly chosen fish has exactly 21 vertebrae, look at the percent of fish that have 21 vertebrae in the given table, because the percent represents the probability in this context. The probability can be converted to decimal form by dividing by 100.

Part B: Finding the Probability for Less Than or Equal to 22 Vertebrae

To find the probability that a randomly chosen fish has 22 or fewer vertebrae, sum the percentages of fish with 20, 21, and 22 vertebrae and then convert that sum to a decimal form by dividing by 100.

Part C: Finding the Probability for More Than 21 Vertebrae

To find the probability that a randomly chosen fish has more than 21 vertebrae, sum the percentages of fish with more than 21 vertebrae, that is 22 and 23 vertebrae, and convert the result to a decimal form.

Part D: Finding the Probability for No More Than 21 Vertebrae

To find the probability that a randomly chosen fish has 21 or fewer vertebrae, add the percent of fish with 20 vertebrae and the percent with 21 vertebrae together, then convert this sum into probability in decimal form.

Key concepts

Probability Distribution

Probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. In the context of the freshwater sculpin, Cottus rotheus, the distribution of the number of tail vertebrae among the population can be mapped out using a probability distribution table.

This table provides us not only with the specific count of vertebrae but also assigns a probability (in this case, a percentage) to each of these counts. To interpret the table effectively, each percentage figure can be considered as a probability by dividing it by 100. For example, the probability of a sculpin having 21 vertebrae is given as 51%, which means that there is a 0.51 probability (51/100) of randomly selecting a sculpin with 21 vertebrae from the population.

Probability distributions are crucial because they provide a complete picture of the variation within a set of data and are the foundation for statistical analysis and hypothesis testing.

Statistical Analysis

Statistical analysis encompasses methods for collecting, analyzing, interpreting, and presenting empirical data. In probability and statistics, it's the practice of making sense of collected data by using different tools and techniques, such as probability distributions.

For example, in the Cottus rotheus exercise, statistical analysis is performed when summarizing the distributions of tail vertebrae and computing probabilities of different events like the likelihood of finding a fish with a specific number of vertebrae. Understanding and applying statistical analysis allows us to make educated conclusions, as evidenced by calculating the probability for a fish having no more than 21 vertebrae. This is done by summing up the individual probabilities for having 20 and 21 vertebrae respectively, reflecting the cumulative nature of probability in less than or equal to scenarios.

This kind of analysis is not only limited to academic exercises but also plays a fundamental role in a wide range of fields including economics, engineering, medicine, and public policy, helping stakeholders make data-driven decisions.

Random Variable

A random variable is a numerical description of the outcome of a statistical experiment. It is a variable that can take on different values at different times due to chance or the inherent variability of the stochastic process involved.

In our textbook example with the sculpin fish, the number of tail vertebrae that a randomly chosen fish has can be considered a random variable. This is because it's determined by chance from the underlying random process, i.e., the genetic and environmental factors affecting vertebrae development in these fish.

Random variables can be discrete, taking on specific values like the whole numbers of vertebrae, or continuous, able to assume any value within a range. The type of random variable helps determine the appropriate probability distribution. The exercise demonstrates how to compute probabilities associated with a discrete random variable, enhancing a student's understanding of how probabilistic concepts apply to quantifiable traits in biology.