Problem 3 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

At a London underground station, the number of women was counted in each of 100 queues of length 10 . In this way a dataset \(x_{1}, x_{2}, \ldots, x_{100}\) was obtained, where \(x_{i}\) denotes the observed number of women in the \(i\) th queue. The dataset is summarized in the following table and lists the number of queues with 0 women, 1 woman, 2 women, etc. $$ \begin{array}{lrrrrrrrrrrr} \hline \hline \text { Count } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \text { Frequency } & 1 & 3 & 4 & 23 & 25 & 19 & 18 & 5 & 1 & 1 & 0 \\ \hline \hline \end{array} $$ In the statistical model for this dataset, we assume that the observed counts are a realization of a random sample \(X_{1}, X_{2}, \ldots, X_{100}\). a. Assume that people line up in such a way that a man or woman in a certain position is independent of the other positions, and that in each position one has a woman with equal probability. What is an appropriate choice for the model distribution? b. Use the table to find an estimate for the parameter(s) of the model distribution chosen in part a.

Step-by-Step Solution

Answer

The model distribution is binomial with \(n = 10\) and estimated \(p = 0.435\).

Step 1: Identify the Model Distribution

Given that each position in the queue is independent and has an equal probability of being a woman or a man, we assume a binomial distribution. Each queue has 10 positions, and thus the problem can be modeled as a binomial distribution with parameters \(n = 10\) and \(p\), where \(p\) is the probability of a woman being in one position.

Step 2: Calculate the Sample Mean

To estimate the probability \(p\), we first calculate the sample mean of the women across all queues. Use the given frequencies to compute this. The mean \(\bar{x}\) is calculated as: \[ \bar{x} = \frac{1(0) + 3(1) + 4(2) + 23(3) + 25(4) + 19(5) + 18(6) + 5(7) + 1(8) + 1(9) + 0(10)}{100} \]

Step 3: Solve for the Sample Mean

Calculate \(\bar{x}\) using the frequency data: \[ \bar{x} = \frac{0 + 3 + 8 + 69 + 100 + 95 + 108 + 35 + 8 + 9 + 0}{100} \] Simplifying gives \(\bar{x} = \frac{435}{100} = 4.35\).

Step 4: Estimate the Parameter \(p\)

The sample mean of a binomial distribution \(X \sim B(n, p)\) is given by \(np\). Since \(n = 10\), solve for \(p\): \[ 10p = 4.35 \]\[ p = \frac{4.35}{10} = 0.435 \]

Key Concepts

Probability Estimation

The process of probability estimation seeks to determine how likely certain outcomes are to occur. In our problem, we estimate the probability of a woman being in a given position in a queue at a London underground station. We use the sample mean to estimate this probability.

To compute the sample mean, we take into account the frequencies of different numbers of women in queues of 10 people each. By weighting these frequencies, we can estimate the expected number of women in any given queue.

The calculation \[\overline{x} = \frac{435}{100} = 4.35\] shows that, on average, there are about 4.35 women per queue. Since each queue has 10 positions, this data helps us estimate the probability \(p\) of any single position being occupied by a woman. By relating this mean to the structure of a binomial distribution \((np)\), we solve for \(p\), yielding \(p = 0.435\). This means that the probability of any specific position being taken by a woman is 43.5%.

This estimation is crucial for making data-driven decisions, which is a key component of probability estimation.

Random Sampling

Random sampling is a cornerstone of drawing meaningful conclusions from data. It refers to the process where each member of a population has an equal chance of being selected.

In the context of our exercise, the queues at the London underground station represent a random sample from a larger population of queues. Each queue is a snapshot, providing valuable data about the overall population.

The assumption of independence is critical in random sampling. In our example, each position in the queue is considered independent, meaning the gender of the person occupying one position doesn't influence the gender of persons in other positions. This random placement of individuals is what justifies using a random sampling approach.

With a random sample, we can generalize our findings beyond the specific sample to the broader population, assuming our sample is representative. This is why it's so important to ensure randomness in sampling processes, as it minimizes bias and makes findings more reliable.

Statistical Modeling

Statistical modeling involves using mathematics to make sense of data. It is about finding the right framework to explain or predict an outcome from the observed data. In this exercise, we choose a binomial distribution to model the number of women in queues.

A binomial distribution is suitable here because it involves trials that have only two possible outcomes: a particular position in the queue is either occupied by a woman (success) or not (failure). Each queue, having 10 independent positions, fits well with this model as we count the number of 'successes' in each trial (each queue).

Through statistical modeling, we calculate the parameter \(p\) for our binomial distribution, which frames it completely. This approach enables us to make predictions about unobserved data, understand variability, and interpret the random sampling's outcomes in a meaningful way.

Understanding statistical modeling is paramount for anyone engaged in data analysis, as it provides the tools needed to convert raw data into insights.