Use the TI-84 Plus calculator to find the z-scores that bound the middle 82% of the area under the standard normal curve. Enter the answers in ascending and round to two decimal places. The z-scores for the given area are and X
The Correct Answer and Explanation is :
To find the z-scores that bound the middle 82% of the area under the standard normal curve using a TI-84 Plus calculator, you would typically follow these steps:
Step-by-Step Guide:
- Understand the Problem:
- The problem asks for the z-scores that correspond to the middle 82% of the area under the standard normal curve.
- The standard normal curve has a mean of 0 and a standard deviation of 1.
- The area under the curve is symmetric, so the total area to the left and right of the center must add up to 82%. This means that 9% of the area is in each tail (since (100\% – 82\% = 18\%), and (18\%) divided by 2 gives 9% for each tail).
- Use the Inverse Normal Function:
- The area in each tail is 9%, so the cumulative area from the left to the z-score in question is 91% (since 100% – 9% = 91%).
- We will use the inverse normal function to find the z-scores for this cumulative area.
- Using the TI-84 Plus:
- Press the 2nd button and then the VARS button to bring up the distribution menu.
- Select invNorm( ), which is the inverse normal function.
- For the z-score on the left side, enter the cumulative area of 0.09 (9% in the left tail).
- For the z-score on the right side, enter the cumulative area of 0.91 (91% cumulative area from the left tail). The inputs will look like this:
- invNorm(0.09, 0, 1) will give you the z-score for the left tail.
- invNorm(0.91, 0, 1) will give you the z-score for the right tail.
- Interpret the Results:
- The TI-84 Plus will output two z-scores. These z-scores will be the ones that bound the middle 82% of the standard normal curve.
Solution:
After using the TI-84 Plus calculator, the z-scores for the 82% middle area are:
- Left z-score: approximately -1.41
- Right z-score: approximately 1.41
Explanation:
- The middle 82% of the standard normal curve is symmetrically distributed around the mean of 0. Therefore, the two z-scores are equally distant from the mean, with one negative and one positive.
- The 82% area corresponds to 9% in each tail of the curve, which means the z-scores that mark the boundaries of the central 82% area are -1.41 (left) and 1.41 (right).
- These values are based on the inverse normal function, which gives us the z-scores for a specified cumulative probability under the normal curve.
By using these z-scores, you can now understand the bounds for the middle 82% of the data in a normal distribution.