91Ó°ÊÓ

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes. It provides the foundation for understanding events that happen by chance and helps in making predictions. In this exercise, we use probability theory to determine the chance of a certain number of voters in a sample.

One key component is the concept of a **random trial**. Each person surveyed can be considered a trial. Moreover, each trial has a **success probability** which, in this case, is the probability that a person voted. Here, the success probability is 0.61 or 61%.

In probability theory, the **binomial distribution** is commonly used to model situations involving a fixed number of trials, each with the same success probability. It allows us to calculate the chances of obtaining a specific number of successful outcomes, such as the number of people who voted out of 1002 surveyed.

When dealing with a large number of trials, like 1002 in this problem, the calculations using the binomial distribution become cumbersome. To simplify, we use the **normal approximation**. This method allows us to approximate the binomial distribution with a normal distribution when the number of trials (n) is large enough.

To apply this, we first calculate the **mean** and **standard deviation** of the binomial distribution. The mean \( \text{mean} = np \), where n is the number of trials and p is the success probability.

For this exercise:
\[ \text{mean} = 1002 \times 0.61 = 611.22 \]
The standard deviation \( \text{std} = \sqrt{np(1-p)} \).

For this problem:
\[ \text{std} = \sqrt{1002 \times 0.61 \times 0.39} = \sqrt{238.6938} \approx 15.45 \]
Now, we have a normal distribution with a mean of 611.22 and a standard deviation of 15.45. This conversion makes it easier to calculate probabilities for the number of voters.

The **z-score** is a way to describe a data point's position relative to the mean of a group of values, measured in terms of standard deviations. It's essential in converting different distributions to the standard normal distribution, making it easier to find probabilities.

To find the z-score, we use the formula:
\( \text{z} = \frac{x - \text{mean}}{\text{std}} \)
where x is the data point we are interested in.

In this problem, we want to know the z-score for x = 701:
\[ \text{z} = \frac{701 - 611.22}{15.45} = 5.81 \]
A z-score of 5.81 is quite high, meaning that 701 voters is far from the mean of 611.22 voters, making this event very unlikely under the normal distribution.

Statistical inference is the process of making conclusions about a population based on sample data. It's a powerful tool used to make predictions or decisions in uncertain situations.

In this exercise, we aim to infer whether the number of people claiming to have voted (701 out of 1002) is consistent with the actual voting rate (61%).

After normal approximation and calculating the z-score, we find a very low probability of this event happening by chance alone. The exceedingly high z-score (5.81) tells us the event is extremely rare given the true voting rate. Hence, statistical inference suggests that the reported number of voters is highly unlikely, questioning the accuracy of the survey claims.

This technique helps us bridge the gap between raw data and meaningful insights, allowing us to draw conclusions about the real world.