91Ó°ÊÓ

Refer to the accompanying table, which describes results from groups of 8 births from 8 different sets of parents. The random variable \(x\) represents the number of girls among 8 children. $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Number of } \\ \text { Girls } \boldsymbol{x} \end{array} & \boldsymbol{P}(\boldsymbol{x}) \\ \hline 0 & 0.004 \\ \hline 1 & 0.031 \\ \hline 2 & 0.109 \\ \hline 3 & 0.219 \\ \hline 4 & 0.273 \\ \hline 5 & 0.219 \\ \hline 6 & 0.109 \\ \hline 7 & 0.031 \\ \hline 8 & 0.004 \\ \hline \end{array}$$ a. Find the probability of getting exactly 1 girl in 8 births. b. Find the probability of getting 1 or fewer girls in 8 births. c. Which probability is relevant for determining whether 1 is a significantly low number of girls in 8 births: the result from part (a) or part (b)? d. Is 1 a significantly low number of girls in 8 births? Why or why not?

Step by step solution

01

Finding probability of exactly 1 girl

From the table, find the probability corresponding to exactly 1 girl. The probability is directly given as \(P(1) = 0.031\).

02

Finding probability of 1 or fewer girls

Add the probabilities of getting 0 and 1 girls from the table: \(P(0) + P(1) = 0.004 + 0.031 = 0.035\).

03

Identifying relevant probability for significance

To determine if 1 girl is a significantly low number, use the cumulative probability of getting 1 or fewer girls (from part b) as it shows the likelihood of that outcome or more extreme ones.

04

Determining if 1 girl is significantly low

Compare the cumulative probability (from part b) with a significance threshold, often 0.05. Since \(0.035 < 0.05\), 1 girl is considered a significantly low number of girls in 8 births.

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Key Concepts

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

Probability Distribution

In the exercise, we deal with a binomial probability distribution. Here, the random variable \(x\) represents the number of girls among 8 births. A probability distribution lists all possible outcomes of a random variable and the corresponding probabilities for each outcome.
For instance, our table shows various outcomes, from having 0 girls to having 8 girls. Each outcome has an associated probability. The probabilities are crucial as they sum up to 1. This property ensures that one of the outcomes will definitely occur.
Understanding a probability distribution helps us answer specific questions. For example, to find the probability of getting exactly 1 girl, we simply look up \(P(1)\) in the table, which is 0.031. This means there’s a 3.1% chance of having exactly 1 girl out of 8 births.

Cumulative Probability

Cumulative probability helps us determine the probability of a random variable falling within a certain range. It’s the sum of the probabilities of outcomes up to and including a specific value.
In our exercise, calculating the cumulative probability is essential for part (b). To find the probability of getting 1 or fewer girls, we add the probabilities of getting exactly 0 girls and exactly 1 girl. From the table: \(P(0) + P(1) = 0.004 + 0.031 = 0.035\).
This result tells us that there’s a 3.5% chance of having 1 or fewer girls among 8 births. This cumulative probability is valuable especially when assessing the likelihood of multiple events.

Significance Testing

Significance testing helps determine if an observed result is unusual or noteworthy in context. We use it to decide if an outcome is significantly different from what we would expect under normal circumstances.
In part (c) of the exercise, we need to examine whether 1 girl out of 8 births is statistically significant. The relevant probability here is the cumulative probability from part (b), which is 0.035.
We compare this cumulative probability to a threshold value, often set at 0.05. If the probability is below this threshold, the result is considered statistically significant. In our case, since 0.035 < 0.05, having 1 girl is seen as a significantly low number among 8 births.

Random Variables

Random variables represent outcomes of a random process. In the exercise, our random variable \(x\) denotes the number of girls among 8 births. It can take on any integer value from 0 to 8.
Random variables can be discrete or continuous. Here, \(x\) is discrete because it takes on specific values (whole numbers). Understanding random variables helps us assign probabilities and analyze outcomes.
For example, knowing \(x\) can range from 0 to 8, each with a corresponding probability, allows us to answer various questions about the births. Whether we're calculating the likelihood of having a certain number of girls or if an observed result is significant, the concept of random variables is deeply integrated into these analyses.