7.2. Example 2

Today we will focus on an important concept in statistics, the normal distribution. Can anyone tell me what characteristics define a normal distribution?

It has a bell-shaped curve and is symmetric around the mean, right?

Exactly! The normal distribution is symmetric, and its mean, median, and mode are all equal. The area under the curve is 1.0. Let’s remember this using the acronym 'SMA' for Symmetric, Mean=Median=Mode, and Area=1.

That’s helpful! What about the empirical rule?

Good question! The empirical rule states that approximately 68%, 95%, and 99.7% of the data fall within 1, 2, and 3 standard deviations from the mean, respectively. We can think of it as '68-95-99.7' for easy recall.

Next, let’s talk about standard normal distribution. To work with normal distributions more easily, we convert our original scores into z-scores. Who can remind us how to calculate a z-score?

It's Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

Correct! Using z-scores, we can easily look up probabilities. For instance, if we want to find the probability of scoring below a certain threshold, we can simply look up the z-score in the z-table.

Can you show us an example of this?

Of course! In our next example, we will find the critical score to be in the top 5%. We will find the corresponding z-score first, then convert back to the original score.

Let’s apply our concepts by solving an example. We have a test score distribution X ~ N(70, 12). How can we determine the minimum score required for the top 5%?

I think we need to find the z-score for 95%.

Exactly! The z-score for 95% is 1.645. Now, applying this to our formula for x:

So, x = 70 + 1.645 * 12? That gives us 89.74, right?

Great job! So, to be in the top 5%, the minimum score is indeed 89.74.

That’s really useful! This helps show how the normal distribution applies in real scenarios.