The mean of a normal probability distribution is 60; the standard deviation is 5
What is your question?
The mean of a normal probability distribution is 60; the standard deviation is 5.
About what percent of the observations lie between 55 and 65?
About what percent of the observations lie between 50 and 70?
About what percent of the observations lie between 45 and 75?
To find the probability of a specific value or range in a normal distribution, you can use the Z-score formula. The Z-score measures the distance of a data point from the mean in terms of standard deviations.
The Z-score formula is:
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the value you want to find the probability for,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
Let's go through an example.
Example: Find the probability of getting a value less than 65 in a normal distribution with a mean of 60 and a standard deviation of 5.
Step 1: Calculate the Z-score using the formula.
Z = (X - μ) / σ
Z = (65 - 60) / 5
Z = 5 / 5
Z = 1
Step 2: Look up the cumulative probability of the Z-score in the standard normal distribution table. The cumulative probability represents the area under the curve up to a specific Z-score.
The standard normal distribution table provides the cumulative probability for positive Z-scores. In this case, Z = 1 represents the area to the left of the Z-score.
From the table, the cumulative probability for Z = 1 is approximately 0.8413.
Step 3: Interpret the cumulative probability.
The cumulative probability of 0.8413 means that there is an 84.13% chance of getting a value less than 65 in a normal distribution with a mean of 60 and a standard deviation of 5.
To find the mean and standard deviation of a normal probability distribution, you need to analyze the given information.
In this case, the mean of the normal probability distribution is stated as 60, and the standard deviation is given as 5.
The mean (μ) of a normal distribution represents its center or average value. In this case, the mean is 60.
The standard deviation (σ) of a normal distribution measures the spread or dispersion of the data around the mean. A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation signifies a greater spread of data points. In this case, the standard deviation is 5.
Therefore, the mean of this normal probability distribution is 60, and the standard deviation is 5.