91Ó°ÊÓ

The following table of data obtained from www.baseball-almanac.com shows hit information for four players. Suppose that one hit from the table is randomly selected. $$\begin{array}{|l|l|l|l|l|}\hline \text { Name } & {\text { single }} & {\text { Double }} & {\text { Triple }} & {\text { Home Run }} & {\text { Total Hits }} \\ \hline \text { Babe Ruth } & {1,517} & {506} & {136} & {714} & {2,873} \\ \hline \text { Jackie Robinson } & {1,054} & {273} & {54} & {137} & {1,518} \\ \hline \text { Ty Cobb } & {3,603} & {174} & {295} & {114} & {4,189} \\ \hline \text { Hank Aaron } & {2,294} & {624} & {98} & {755} & {3,771} \\ \hline\end{array}$$ Are "the hit being made by Hank Aaron" and "the hit being a double" independent events? a. Yes, because P(hit by Hank Aaron|hit is a double) = P(hit by Hank Aaron) b. No, because P(hit by Hank Aaron|hit is a double) ? P(hit is a double) c. No, because P(hit is by Hank Aaron|hit is a double) ? P(hit by Hank Aaron) d. Yes, because P(hit is by Hank Aaron|hit is a double) = P(hit is a double)

Step by step solution

01

Determine Total Number of Hits

We need to find the total number of hits made by all the players combined. Sum the 'Total Hits' for each player: \(2873 + 1518 + 4189 + 3771 = 12351\).

02

Calculate Probability of Hit by Hank Aaron

Find the probability that a hit was made by Hank Aaron. This is the number of hits by Hank Aaron divided by the total number of hits: \( \frac{3771}{12351} \approx 0.3054 \).

03

Calculate Probability of a Double

Determine the probability that a hit is a double. Sum all the doubles: \(506 + 273 + 174 + 624 = 1577\). Then divide by the total number of hits: \( \frac{1577}{12351} \approx 0.1277 \).

04

Calculate Conditional Probability

Compute the probability that a hit is made by Hank Aaron given that it is a double. This is the number of doubles by Hank Aaron divided by the total number of doubles: \( \frac{624}{1577} \approx 0.3956 \).

05

Compare Probabilities for Independence

Compare \( P(\text{hit by Hank Aaron | hit is a double}) \approx 0.3956 \) with \( P(\text{hit by Hank Aaron}) \approx 0.3054 \). Since these probabilities are not equal, the events are not independent.

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Key Concepts

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

Probability

Probability is a fundamental concept in statistics and is all about measuring how likely an event is to happen. Imagine you're throwing a ball into a basket. If you manage to do this 3 times out of 10, the probability is 0.3 or 30%.
Probabilities are always numbers between 0 and 1. Zero means the event will never occur, while one means it100% will. We express these numbers in fractions, decimals, or percentages.
Let's dive into the baseball example. We wanted to know the probability that a hit was made by Hank Aaron. To find this, we took the number of hits Hank Aaron made and divided it by the total number of hits. This gives us the probability as a fraction or decimal, such as 0.3054. This means there's about a 30.54% chance of a hit being made by him if you randomly select a hit from the list.

Independence of Events

Understanding if events are independent is crucial in probability, as it shows whether the occurrence of one event affects another. Imagine rolling dice; the result of one roll doesn't impact the next, making these events independent. However, the color of a card drawn may depend on whether certain others have been drawn already. This would make them dependent.
In our baseball exercise, we wanted to know if whether a hit was made by Hank Aaron was independent of the hit being a double. To assess independence, we need to compare two probabilities:

• Conditional Probability: The chance of Hank Aaron making a hit given it is a double.
• Probability of Hank Aaron making any hit.

If these two probabilities are the same, then the two events are independent. If they differ, like in our exercise, they are dependent. Here, the probabilities weren't equal, meaning the specific type of hit impacts the likelihood of Hank Aaron being the batter.

Statistical Analysis

Statistical analysis involves using statistical methods to interpret data and draw conclusions. It's like being a detective, searching for patterns and hidden relationships in data. Complex statistical tools and tests help us do this, but we often start with simple tools like sums and averages.
In this exercise, we conducted a straightforward statistical analysis by gathering data on hits from four baseball players. We calculated total hits and types of hits for comparisons. By doing this, we could determine probabilities and conditional probabilities to draw conclusions about independence.
The most notable analysis came through the comparison of probabilities. It revealed that some baseball hitting events, like those involving Hank Aaron, weren't independent, providing a deeper insight into the data we had. Remember, statistical analysis is not just about numbers; it's about making sense of what they mean and how they relate.