Hey guys! Let's dive into the fascinating world of normal distribution and tackle some probability problems. Normal distribution, often called the Gaussian distribution, is a cornerstone of statistics. It's characterized by its bell-shaped curve, which is symmetrical around the mean. In this article, we're going to calculate probabilities associated with a random variable z that follows a standard normal distribution, meaning it has a mean () of 0 and a standard deviation () of 1. We'll be using z-scores and looking them up in a standard normal distribution table (or using software) to find these probabilities. So, buckle up, and let's get started!
Delving into Normal Distribution and Probability
Before we jump into specific problems, let's solidify our understanding of what probabilities in a normal distribution actually mean. In the context of a standard normal distribution (with and ), the z-score represents the number of standard deviations a particular value is away from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it's below the mean. The standard normal distribution table (also known as the z-table) provides the cumulative probability, which is the probability that a random variable z is less than or equal to a specific z-score. This is a crucial concept, as many probability calculations rely on finding these cumulative probabilities.
To calculate probabilities such as P(z < a), we directly look up the z-score 'a' in the z-table. For probabilities like P(z > a), we use the property that the total probability under the normal curve is 1, so P(z > a) = 1 - P(z ≤ a). When we need to find the probability between two z-scores, P(a < z < b), we calculate the difference between the cumulative probabilities: P(a < z < b) = P(z < b) - P(z < a). These basic rules and understandings are essential for solving a wide range of probability problems related to the normal distribution. Moreover, understanding these probabilities helps us in real-world applications such as hypothesis testing, confidence interval estimation, and understanding data distributions in various fields, from finance to engineering. So, keep these concepts in mind as we solve the specific probability questions below.
(a) Calculating P(-0.87 < z < 1.28)
Let's start with our first problem: determining the probability that our random variable z falls between -0.87 and 1.28, mathematically expressed as P(-0.87 < z < 1.28). To solve this, we'll use the cumulative probability concept we discussed earlier. We need to find the area under the standard normal curve between these two z-scores. Here's the breakdown:
First, we find the cumulative probability for z = 1.28, which is P(z < 1.28). This value represents the area under the curve to the left of z = 1.28. Looking up 1.28 in a standard normal distribution table (or using software), we find this probability to be approximately 0.8997. This means there's an 89.97% chance that z will be less than 1.28.
Next, we find the cumulative probability for z = -0.87, which is P(z < -0.87). This value represents the area under the curve to the left of z = -0.87. Looking up -0.87 in the z-table, we find this probability to be approximately 0.1922. So, there's a 19.22% chance that z will be less than -0.87.
Now, to find the probability that z lies between -0.87 and 1.28, we subtract the cumulative probability of the lower bound (-0.87) from the cumulative probability of the upper bound (1.28). This gives us P(-0.87 < z < 1.28) = P(z < 1.28) - P(z < -0.87) = 0.8997 - 0.1922 = 0.7075. Therefore, the probability that z falls between -0.87 and 1.28 is approximately 0.7075. In layman's terms, there's about a 70.75% chance that a randomly selected value from this standard normal distribution will fall within this range. This calculation highlights how cumulative probabilities are used to find probabilities within intervals, a fundamental skill in statistical analysis.
(b) Determining P(-0.50 < z < 1.00)
Moving on to our second probability calculation, we aim to determine P(-0.50 < z < 1.00). This is similar to the previous problem, where we need to find the probability that the random variable z falls within a specific range. Again, we'll leverage the cumulative probabilities from the standard normal distribution table.
First, let's find the cumulative probability for z = 1.00, denoted as P(z < 1.00). This represents the area under the standard normal curve to the left of z = 1.00. Consulting the z-table, we find that P(z < 1.00) is approximately 0.8413. This tells us that about 84.13% of the distribution lies below z = 1.00.
Next, we determine the cumulative probability for z = -0.50, or P(z < -0.50). This is the area under the curve to the left of z = -0.50. Looking up -0.50 in the z-table, we find the probability to be approximately 0.3085. Thus, about 30.85% of the distribution is below z = -0.50.
To find the probability P(-0.50 < z < 1.00), we subtract the cumulative probability of the lower bound (z = -0.50) from the cumulative probability of the upper bound (z = 1.00). This gives us P(-0.50 < z < 1.00) = P(z < 1.00) - P(z < -0.50) = 0.8413 - 0.3085 = 0.5328. Therefore, the probability that z falls between -0.50 and 1.00 is approximately 0.5328. This means there's a 53.28% chance that a randomly chosen value from the standard normal distribution will fall within this interval. This problem reinforces the method of using cumulative probabilities to find the probabilities within a range, a vital skill in statistical analysis and probability calculations.
(c) Calculating P(z > -1.35)
Now, let's tackle the probability P(z > -1.35). This is slightly different from the previous examples because we're looking for the probability that z is greater than a certain value, rather than less than or between values. However, we can still use the cumulative probabilities and a key property of the normal distribution to find the solution.
Remember that the total area under the standard normal curve is equal to 1, representing 100% probability. The cumulative probability, which we find in the z-table, gives us the probability that z is less than or equal to a particular value. So, to find the probability that z is greater than -1.35, we can subtract the probability that z is less than or equal to -1.35 from 1.
Mathematically, this is expressed as P(z > -1.35) = 1 - P(z ≤ -1.35). First, we need to find P(z ≤ -1.35). Consulting the standard normal distribution table, we look up the z-score -1.35 and find its cumulative probability to be approximately 0.0885. This means there's an 8.85% chance that z will be less than or equal to -1.35.
Now, we subtract this value from 1 to find the probability that z is greater than -1.35: P(z > -1.35) = 1 - 0.0885 = 0.9115. Therefore, the probability that z is greater than -1.35 is approximately 0.9115. In other words, there's a 91.15% chance that a randomly selected value from the standard normal distribution will be greater than -1.35. This calculation demonstrates a common technique for finding probabilities in the upper tail of the distribution, where we use the complement rule to relate 'greater than' probabilities to cumulative probabilities.
(d) Determining P(z < -1.40 or z > 1.62)
Finally, let's tackle the last probability calculation: P(z < -1.40 or z > 1.62). This problem involves finding the probability of two separate events and combining them since they are connected by the word
