Final answer:
The topic discussed is the standard normal distribution, focusing on calculating z-scores and using them to find probabilities for normally distributed variables. The Central Limit Theorem also allows for the application of this concept to sample means.
Step-by-step explanation:
Understanding the Standard Normal Distribution
The question relates to the standard normal distribution, an important concept in statistics. When variable X is normally distributed with a mean (μ) of 10 and an unknown standard deviation (σ), we can still discuss its properties and calculate probabilities. A z-score represents how many standard deviations a particular value of X is from the mean. The distribution of z-scores is known as the Standard Normal Distribution, denoted as Z ~ N(0, 1), which has a mean of 0 and a standard deviation of 1.
To calculate the probability associated with a certain value of X, you would use the z-score formula z = (X - μ) / σ. Once the z-score is calculated, you can find the corresponding probability by referring to a z-table or using a statistical calculator to compute the area under the normal curve. This area under the curve corresponds to the probability of observing a value less than or equal to X.
When applying this to sample means, the Central Limit Theorem suggests that the distribution of sample means will approximate a normal distribution, even if the underlying distribution is not normal, provided the sample size is sufficiently large. This means that we can use z-scores and the standard normal distribution to calculate probabilities and make inferences about population means based on sample data.