Probability Calculation: Sample Proportion Between 0.37 & 0.39

Hey guys! Let's dive into a probability problem where we need to figure out the likelihood of a sample proportion falling within a specific range. We'll break it down step-by-step, so it's super easy to follow. We're given a mean, a standard deviation, and a range, and our mission is to find the probability. So, grab your thinking caps, and let's get started!

Understanding the Problem

So, the core of this problem revolves around calculating probability. Probability is essentially the measure of the likelihood that an event will occur. It's a fundamental concept in statistics and helps us make predictions and informed decisions based on data. In our specific scenario, we’re dealing with a sample proportion, which represents the fraction of a sample that possesses a certain characteristic. In simpler terms, imagine we're surveying a group of registered voters. Our sample proportion would be the fraction of those surveyed who actually voted. The problem gives us a mean (0.38) and a standard deviation (0.0485), which are crucial pieces of information. The mean, or average, gives us a central point around which our data clusters. The standard deviation, on the other hand, tells us how spread out the data is from this mean. A smaller standard deviation means the data points are closer to the mean, while a larger one indicates they're more dispersed. Our goal is to determine the probability that a randomly chosen sample has a proportion of voters between 0.37 and 0.39. This means we want to know how likely it is to find a sample where the voting proportion falls within this specific range, given the mean and standard deviation we already have. To tackle this, we'll need to use some statistical tools and concepts, which we'll explore in detail in the following sections. Understanding these foundational elements is key to solving the problem accurately and confidently, so let’s move on and see how we can apply them.

Key Concepts: Normal Distribution and Z-Scores

Before we jump into calculations, let’s talk about the two big concepts that'll help us solve this: the normal distribution and z-scores. Think of the normal distribution as a bell curve – it's symmetrical, with the peak in the middle representing the mean. Many natural phenomena, like heights or test scores, tend to follow this distribution. When we're dealing with sample proportions, the distribution of those proportions also often approximates a normal distribution, especially when the sample size is large enough. This is super helpful because it means we can use the properties of the normal distribution to calculate probabilities. Now, let's throw z-scores into the mix. A z-score tells us how many standard deviations a particular value is away from the mean. It's like a standardized ruler that helps us compare values from different normal distributions. For instance, a z-score of 1 means the value is one standard deviation above the mean, while a z-score of -1 means it's one standard deviation below the mean. Calculating z-scores is crucial because it allows us to use the standard normal table (or a calculator) to find probabilities. The standard normal table gives us the area under the curve to the left of a given z-score, which corresponds to the cumulative probability up to that point. So, by converting our sample proportions (0.37 and 0.39) into z-scores, we can then use the standard normal table to find the probabilities associated with those z-scores. This will ultimately help us determine the probability that the sample proportion falls between 0.37 and 0.39. Knowing these concepts inside and out is like having the secret decoder ring for this problem, so let’s move on to the next step: calculating those essential z-scores!

Calculating Z-Scores

Alright, guys, let's get down to the nitty-gritty and calculate some z-scores! As we discussed, z-scores are crucial for figuring out how many standard deviations our values are from the mean. The formula for calculating a z-score is pretty straightforward: z = (x - μ) / σ, where 'x' is the value we're interested in, 'μ' is the mean, and 'σ' is the standard deviation. In our problem, we have a mean (μ) of 0.38 and a standard deviation (σ) of 0.0485. We want to find the probability that a sample proportion falls between 0.37 and 0.39, so we need to calculate z-scores for both of these values. First, let's calculate the z-score for 0.37. Plugging the values into our formula, we get: z1 = (0.37 - 0.38) / 0.0485. This simplifies to z1 = -0.01 / 0.0485, which gives us a z-score of approximately -0.206. This tells us that 0.37 is about 0.206 standard deviations below the mean. Now, let's do the same for 0.39. The formula becomes: z2 = (0.39 - 0.38) / 0.0485. This simplifies to z2 = 0.01 / 0.0485, resulting in a z-score of approximately 0.206. This means that 0.39 is about 0.206 standard deviations above the mean. Now that we have our z-scores, we can use these values to find the corresponding probabilities from the standard normal table. These z-scores are our key to unlocking the probabilities we're after, so let's head on to the next step and learn how to use them effectively!

Using the Standard Normal Table

Okay, now comes the fun part: using the standard normal table! Think of this table as our translator – it converts z-scores into probabilities. The standard normal table, also known as the Z-table, shows the cumulative probability associated with a given z-score. This cumulative probability represents the area under the standard normal curve to the left of that z-score. In simpler terms, it tells us the probability of getting a value less than or equal to the value corresponding to the z-score. Remember, we calculated z-scores of approximately -0.206 and 0.206 for 0.37 and 0.39, respectively. To find the probabilities, we need to look up these z-scores in the standard normal table. When you look up -0.206 in the table, you'll find a probability of approximately 0.4187. This means there's about a 41.87% chance of getting a value less than 0.37. Similarly, when you look up 0.206 in the table, you'll find a probability of approximately 0.5813. This indicates that there's about a 58.13% chance of getting a value less than 0.39. But hold on, we're not quite there yet! We want to find the probability that the sample proportion falls between 0.37 and 0.39. This means we need to find the area under the curve between our two z-scores. To do this, we'll subtract the probability associated with the lower z-score from the probability associated with the higher z-score. This will give us the probability of a sample proportion falling within our desired range. So, let's move on to the final calculation and see what the probability is!

Calculating the Final Probability

Alright, time to put it all together and calculate the final probability! We've done the groundwork – calculated the z-scores, and looked up the corresponding probabilities in the standard normal table. Now, we just need to do a little subtraction to find the probability that a sample proportion falls between 0.37 and 0.39. Remember, we found that the probability of getting a value less than 0.37 (z-score of -0.206) is approximately 0.4187, and the probability of getting a value less than 0.39 (z-score of 0.206) is approximately 0.5813. To find the probability of a value falling between 0.37 and 0.39, we subtract the lower probability from the higher probability. So, our calculation looks like this: P(0.37 < proportion < 0.39) = P(z < 0.206) - P(z < -0.206) = 0.5813 - 0.4187. When we do the subtraction, we get 0.1626. This means there's approximately a 16.26% probability that a sample chosen at random will have a proportion of registered voters who vote between 0.37 and 0.39. And there you have it! We've successfully calculated the probability by breaking the problem down into manageable steps, from understanding the concepts of normal distribution and z-scores to using the standard normal table and performing the final calculation. This problem highlights the power of statistical tools in helping us understand and predict real-world scenarios. So, let’s wrap up what we've learned and solidify our understanding.

Conclusion

Okay, guys, let's wrap things up! We've tackled a probability problem involving sample proportions, and hopefully, you now feel like statistical superheroes! We started with a seemingly complex question: