Unveiling Z: Solving Standard Normal Probabilities

Hey math enthusiasts! Today, we're diving into the fascinating world of the standard normal distribution and tackling some probability problems. Our mission? To find specific values of the standard normal random variable, often denoted as z, that satisfy certain probability conditions. Think of z as a point on the famous bell curve. We're essentially trying to pinpoint where this point sits based on the area under the curve. Remember, in a standard normal distribution, the total area under the curve is always 1, and it's perfectly symmetrical around the mean (which is 0 in this case).

Let's get started. We'll break down each problem step-by-step, explaining the concepts and providing the solutions. This is where we learn to read the normal distribution table or use statistical software to find the z-score. Get ready to flex those probability muscles!

a. Finding z₀ Where P(z ≤ z₀) = 0.0344

Alright, guys, let's start with the first problem: finding the value of z₀ where the probability of z being less than or equal to z₀ is 0.0344, i.e., P(z ≤ z₀) = 0.0344. This means we're looking for a z-score such that the area under the standard normal curve to the left of z₀ is 0.0344. The visual representation is the area under the bell curve on the left side, representing 3.44% of the total area. It is important to remember that since the normal distribution is symmetric, values to the left of the mean (0) are negative, and values to the right are positive. Now, we use the standard normal table (also known as the z-table) or a statistical calculator. If you're using a z-table, you'll look for the area closest to 0.0344 within the table. This table gives you areas corresponding to different z-scores. A statistical calculator can give you the exact z-score. In this case, we find a z₀ value of approximately -1.82. Therefore, when z₀ is -1.82, the probability P(z ≤ z₀) is roughly 0.0344. That's our answer. So, the value of z₀ is negative, indicating it's located to the left of the mean on the bell curve. This also makes sense because the probability (0.0344) is less than 0.5, implying that z₀ is less than the mean (0).

Remember, the z-table provides probabilities associated with z-scores, essentially converting z-scores into probabilities, and it is the reverse for this type of problem where we look for the z-score from probability. Understanding how to use the z-table is crucial for tackling these types of problems. Statistical software often provides this information as well, and it can be used to cross-reference our table calculations. Always ensure you're using the correct table (one that gives the area to the left or right, depending on how the problem is framed). A correct reading of the table will give you an accurate z-score.

e. Finding z₀ Where P(-z₀ ≤ z ≤ 0) = 0.3411

Now, let's switch gears and approach the second problem: finding the value of z₀ where P(-z₀ ≤ z ≤ 0) = 0.3411. This question is asking us to find the z-score such that the area under the standard normal curve between -z₀ and 0 (the mean) is 0.3411. Because the standard normal distribution is symmetrical around 0, and because we are dealing with area from -z₀ to 0, which is exactly the area from 0 to z₀, we can infer that the probability from 0 to z₀ is also 0.3411. This means that if we visualize the area, we're looking at the portion of the curve between -z₀ and 0. You can visualize this on the bell curve. The area spans from the negative side to the center (0), representing 34.11% of the total area under the curve.

Now, let's use the z-table to find the corresponding z-score that gives us an area of 0.3411. Remember that the z-table usually provides the area from the mean (0) to a specific z-score. Looking up the area closest to 0.3411 on the z-table, we find a z-score of approximately 1.00. This is because the area from 0 to 1.00 is approximately 0.3413, close to what we need, which makes this problem simpler since we can directly use the z-table. Since the normal distribution is symmetric, we can conclude that the z-score is 1.00. Therefore, z₀ is 1.00 because the area from -1.00 to 0 is 0.3411. Understanding symmetry is key in this problem! Because the standard normal distribution is symmetric, the area under the curve from -z₀ to 0 is the same as the area from 0 to z₀. Knowing this symmetry simplifies the problem and allows you to find the value of z₀ quickly. Always be aware of the characteristics of the standard normal distribution as you work on this type of question. Knowing the symmetry and properties will help you simplify complex calculations. Also, be sure to always verify your results with a statistical calculator or software to ensure accuracy.

b. Finding z₀ Where P(-z₀ ≤ z ≤ z₀) = 0.9544

Alright, let's tackle the next one: finding the value of z₀ where P(-z₀ ≤ z ≤ z₀) = 0.9544. This is another classic problem that requires us to find a z-score. Here, we're looking for a z₀ such that the area under the standard normal curve between -z₀ and z₀ is 0.9544. This means we're looking at the area under the bell curve symmetrically around the mean (0). So, we can divide the total area (0.9544) by 2, which gives us 0.4772 for the area on each side of the mean, from 0 to z₀ and from -z₀ to 0. It is necessary to understand this problem because it is symmetrical around the mean, which is 0.

Now, we'll use the z-table. If you look at the z-table and find the z-score corresponding to an area of 0.4772 (or very close), you will find the z-score approximately equal to 2.00. This means the area under the curve from 0 to 2.00 is roughly 0.4772, and because the distribution is symmetric, the area from -2.00 to 0 is also 0.4772. So, the total area from -2.00 to 2.00 is 0.9544. So, the value of z₀ is 2.00. We can also arrive at the answer because the question indicates that the probability covers the region between -z₀ and z₀, indicating a symmetrical interval. Therefore, we can find the area to the right of 0 and find the corresponding z-score. Knowing the symmetry is fundamental to solving this type of problem. The z-table is a valuable tool, but always double-check your work using a statistical calculator or software to ensure accuracy. This will help you verify your solution and improve your understanding of the standard normal distribution. Also, always make sure you're using the correct z-table – the area either to the left or right of the z-score.

c. Finding z₀ Where P(z ≥ z₀) = 0.0228

Let's keep the ball rolling with the next problem: finding z₀ where P(z ≥ z₀) = 0.0228. In this instance, we are looking for the z-score such that the area under the standard normal curve to the right of z₀ is 0.0228. This means the probability of z being greater than or equal to z₀ is 0.0228. The area is on the right side of the bell curve, a small part because the probability is small, representing 2.28% of the total area. Remember that the total area under the curve is always 1. Thus, the area to the left of z₀ is 1 - 0.0228 = 0.9772.

Now, use the z-table to find the z-score for the area that is approximately 0.9772. When you look up the value of 0.9772 on the z-table, you'll find that it corresponds to a z-score of approximately 2.00. This is because the table typically provides areas from the mean to the z-score. But since this problem is the area to the right, we subtract the value from 1 to find the z-score. So, z₀ is approximately 2.00. Another approach is to think in terms of the area on the left, but we can also think of the area on the right, which is the same as the value of the table. So, the area on the right is 0.0228, which corresponds to the same z-score, 2.00. Always make sure you understand whether the table gives you the area to the left or right of the z-score to avoid confusion. Using the complement rule helps a lot in understanding this type of probability problem. Always remember that the total area under the curve is equal to 1. This helps in understanding the areas on both sides of z₀. Double-check your answers using statistical software or calculators to make sure you have the correct values.

d. Finding z₀ Where P(z ≥ z₀) = 0.9772

Alright, let's solve the last problem: finding the value of z₀ where P(z ≥ z₀) = 0.9772. In this scenario, we are seeking a z-score such that the area under the standard normal curve to the right of z₀ is 0.9772. This indicates that the probability of z being greater than or equal to z₀ is 0.9772. The area is large on the right side of the bell curve because of the high probability, representing 97.72% of the total area. From this, we can calculate the area on the left of z₀, which is 1 - 0.9772 = 0.0228.

Now, let's use the z-table and look for the z-score associated with an area that is approximately 0.0228. If you look up the area of 0.0228 on the z-table, you'll find that the corresponding z-score is approximately -2.00. This is because the area to the left of -2.00 is 0.0228. So, the area to the right of -2.00 is 0.9772. Thus, z₀ is approximately -2.00. So we can conclude that z₀ must be negative. It is important to remember that since we are dealing with the area to the right, we're looking for the left area and looking up the corresponding z-score. The complement rule is very handy in these situations. Double-check your results with statistical software or calculators. It will improve your understanding of the standard normal distribution and probability problems.

And there you have it, guys! We have successfully tackled these probability problems and found the respective values of z₀. Keep practicing and you'll become a pro at these problems in no time. If you have any questions, feel free to ask! Remember to always double-check your work and use a z-table or statistical calculator to verify your answers. Happy calculating, and keep exploring the amazing world of mathematics!"