Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15.
What is your question?
Yes, it would be unusal for the mean of a sample of 3 to be 115 or more because there is only a 4.18% chance for this to happen.
P(X >= 115)
= P(X >= (115 - 100) / sqrt(15^2 / 3))
= P(X >= 1.732050808)
= 0.0418
Sure, let's assume we have a population that is normally distributed with a mean of 100 and a standard deviation of 15. This means that the data points in this population will follow a bell-shaped curve, with most values clustered around the mean of 100, and the spread of the data extending in both directions based on the standard deviation of 15.
Now, let's explore a few potential questions we can answer based on this information.
1. What is the probability that a randomly selected individual from this population has a value less than 90?
To find this probability, we can use the concept of the standard normal distribution. By standardizing the data using the formula (X - μ) / σ, where X is the value we are interested in, μ is the mean, and σ is the standard deviation, we can then use a Z-table or a statistical software to find the corresponding probability. In this case, (90 - 100) / 15 gives us a Z-score of -0.67. By referring to a Z-table or using a statistical software, we can find that the probability of getting a value less than -0.67 from a standard normal distribution is approximately 0.2514, or 25.14%.
2. What is the probability that a randomly selected individual from this population has a value between 80 and 120?
Similarly, we can use the concept of the standard normal distribution to find this probability. By standardizing the values 80 and 120, we get Z-scores of -1.33 and 1 respectively. Using a Z-table or statistical software, we can find the area under the curve between -1.33 and 1. This probability represents the percentage of data points falling between these two values.
3. What is the value that separates the top 10% from the remaining values in this population?
To find this value, we need to determine the Z-score that corresponds to the top 10% of the data. Using a Z-table or a statistical software, we can find the Z-score such that the area to the right of it is 0.10. This Z-score will correspond to the value separating the top 10% from the remaining 90% of the data. By multiplying this Z-score by the standard deviation and adding it to the mean, we can find the value in the original units that corresponds to the desired probability.
These are just a few of the questions you can answer based on the given information about a normally distributed population. By understanding the properties of the normal distribution and using appropriate statistical techniques, you can explore many more questions and analyze various aspects of the data.