find the probability p(-1.14 < z < 1.01)
using the normal standard distribution
Look up a standard normal distribution table, or use the following link:
http://www.math.unb.ca/~knight/utility/NormTble.htm
The left-tail probability
P1=P(< -1.14σ) = 0.1271
and the left-tail probability
P2=P(< +1.01σ) = 0.8438
So
P(-1.14σ < z < 1.01σ)=P2-P1
Well, if we're talking about the standard normal distribution, we're basically looking for the probability that a randomly selected value falls between -1.14 and 1.01.
Now, let me put on my math clown nose and do some calculations...
To find this probability, we need to use a standard normal distribution table or a calculator. The z-score for -1.14 is approximately -0.369 and the z-score for 1.01 is approximately 0.344.
So, we want to find the probability of getting a z-score between -0.369 and 0.344, which can be written as P(-0.369 < z < 0.344).
Drumroll, please...
The probability of that happening is roughly 0.3557, or about 35.57%.
Keep in mind, my clown calculations could have a margin of error. So, don't bet your whole act on it!
To find the probability, you need to calculate the area under the curve of the standard normal distribution between the values -1.14 and 1.01. Here are the steps to find it:
Step 1: Look up the z-score for -1.14 and 1.01 in the z-table.
The z-score for -1.14 is -0.3693, and the z-score for 1.01 is 0.8442.
Step 2: Calculate the probability using the z-scores.
The probability of -1.14 < z < 1.01 can be calculated by finding the area under the curve between these z-scores.
Probability = P(-1.14 < z < 1.01) = P(z < 1.01) - P(z < -1.14)
Step 3: Look up the probabilities in the standard normal distribution table.
Using the z-values from Step 1, look up the probabilities for P(z < 1.01) and P(z < -1.14) in the standard normal distribution table.
The value for P(z < 1.01) is 0.8438, and the value for P(z < -1.14) is 0.1271.
Step 4: Calculate the final probability.
Subtract the probability for P(z < -1.14) from the probability for P(z < 1.01) to find the final probability.
Probability = P(-1.14 < z < 1.01) = P(z < 1.01) - P(z < -1.14)
= 0.8438 - 0.1271
= 0.7167
Therefore, the probability p(-1.14 < z < 1.01) is approximately 0.7167.
To find the probability P(-1.14 < Z < 1.01) using the normal standard distribution, you can use a standard normal distribution table or a calculator that can calculate the cumulative probability for the standard normal distribution. Here's how you can do it step by step:
Step 1: Understand the notation
The notation Z represents a standard normal random variable. The standard normal distribution has a mean of 0 and a standard deviation of 1.
Step 2: Standardize the values
To use the standard normal distribution, we need to standardize the given values by subtracting the mean and dividing by the standard deviation. In this case, since we are working with a standard normal distribution, the mean (μ) is 0 and the standard deviation (σ) is 1.
So we have:
-1.14 = (x - μ) / σ = (x - 0) / 1 = x
1.01 = (y - μ) / σ = (y - 0) / 1 = y
Step 3: Look up the probabilities
Now that we have standardized our values, we can find the probabilities associated with these values using a standard normal distribution table or a calculator.
The probability P(-1.14 < Z < 1.01) represents the area under the standard normal curve between -1.14 and 1.01. This can be interpreted as the probability of Z falling between -1.14 and 1.01 standard deviations from the mean.
Using a standard normal distribution table or a calculator, you can find the cumulative probabilities associated with the standardized values -1.14 and 1.01. The difference between these two cumulative probabilities will give you the desired probability.
Note: The exact value of the probability may vary depending on the level of precision used in the table or calculator.
Therefore, to find the probability P(-1.14 < Z < 1.01) using the normal standard distribution, you need to use a standard normal distribution table or a calculator to determine the cumulative probabilities associated with these standardized values and subtract one cumulative probability from the other.