Find two z-scores so that 40% of the data in the distribution lies between them. (More than one answer is possible). Include a sketch of the distribution with 40% of the area shaded and the corresponding z-scores included.
I'm not sure how to do this problem. Just need a boost to get started. Please.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
(.40) and the related Z scores.
Or you could use this source:
http://davidmlane.com/hyperstat/z_table.html
To find the z-scores that correspond to a specific percentage of data in a standard normal distribution, we can use the standard normal distribution table or a calculator. The standard normal distribution table provides the area under the standard normal curve from the left-hand side up to a given z-score.
To find two z-scores with 40% of the data in between them, we need to find the z-score that corresponds to the area of 0.40/2 on each side of the distribution curve.
The table provides area values from 0 to 0.50, so we can look for the area closest to 0.20 (0.40/2).
Look for the closest area value of 0.20 in the standard normal distribution table. The corresponding z-score will give us the mean z-score for our desired area.
The sketch of the distribution with 40% of the area shaded would look like this:
```
|-----| |-----|
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| |----| |
|<---1.96--->|
| |
|<-------------->|
-3.09 3.09
```
Hope this helps to get you started!
To find the z-scores such that 40% of the data lies between them, we can follow these steps:
1. Identify the desired area: Since we want 40% of the data to lie between the two z-scores, we know that the total area to the left of the first z-score and the total area to the right of the second z-score should be (100% - 40%) / 2 = 30%.
2. Determine the corresponding z-scores: We can use a standard normal distribution table (also known as a z-table) to find the z-scores that correspond to an area of 30% to the left and 30% to the right.
Alternatively, we can use technology (such as statistical software or an online calculator) to find the z-scores. For instance, we could use the inverse normal function on a calculator to find the z-scores.
Note: Since more than one answer is possible, we will find two pairs of z-scores.
Now, let's sketch the distribution and find the corresponding z-scores.
_______________________
| |
|__________ :___________|
In the sketch above, the shaded portion represents 40% of the area under the curve.
To find the z-scores, we need to determine the z-score corresponding to 30% to the left and the z-score corresponding to 30% to the right.
Using a z-table or calculator, we find that the z-score corresponding to an area of 30% to the left is approximately -0.524. This means that 30% of the data lies to the left of a z-score of -0.524.
Similarly, the z-score corresponding to an area of 30% to the right is approximately 0.524. This implies that 30% of the data lies to the right of a z-score of 0.524.
Therefore, two pairs of possible z-scores that satisfy this condition are:
1) -0.524 and 0.524
2) -0.524 and -0.524
Keep in mind that z-scores can be positive or negative, so either pair is valid.