Find the z-scores for which 92
%
of the distribution's area lies between minus
z
and z.
Well, well, well, if it isn't a z-score question! Let me put on my funny hat and help you out.
To find the z-scores for which 92% of the distribution's area lies between minus z and z, we need to find the corresponding z-scores. This means we're looking for the values of z that cut off 4% of the distribution on each tail (since 100% - 92% = 8%, and we have to divide that by 2 for each tail).
Now, I'm not saying this is a clownish distribution, but let's just imagine it's a bunch of clowns forming a line. Picture it. In order for only 4% of the clowns to be on each end, the rest of the 92% must be squeezed together in the middle like a clown car. Talk about a funny balancing act!
But I digress. We can consult our trusty clown calculator, or rather, a normal distribution table, to find the z-scores that correspond to the tails containing 4% each. And the answer is... (drumroll, please) approximately -1.750 and 1.750!
So, if 92% of the distribution's area lies between minus z and z, the corresponding z-scores are approximately -1.750 and 1.750. Keep in mind that these are just estimates, but hey, clowns are all about having a good time, right?!
To find the z-scores for which 92% of the distribution's area lies between minus z and z, we need to find the z-scores that correspond to the 4% and 96% percentiles.
Step 1: Convert the given percentage to a decimal.
92% = 0.92
Step 2: Find the z-scores that correspond to the 4% and 96% percentiles using a standard normal distribution table or a calculator. The z-score for the 4% percentile is approximately -1.750 and the z-score for the 96% percentile is approximately 1.750.
Thus, the z-scores for which 92% of the distribution's area lies between minus z and z are approximately -1.750 and 1.750.
To find the z-scores for which 92% of the distribution's area lies between minus z and z, you can use the standard normal distribution table or a statistical calculator.
Here's how you can use the standard normal distribution table:
Step 1: Determine the proportion of the area outside the range. In this case, the proportion outside of the range between minus z and z is 1 - 0.92 = 0.08.
Step 2: Divide the proportion by 2 to find the area in just one tail. In this case, 0.08 / 2 = 0.04.
Step 3: Find the corresponding z-scores for the area in just one tail by looking up the value 0.04 in the standard normal distribution table. The table will give you the z-score associated with the cumulative probability of 0.04.
Step 4: Since we are interested in the range between minus z and z, we need to find the z-score that has a cumulative probability of 0.04. This z-score will be the negative value of the z-score obtained in Step 3.
For example, if the z-score obtained in Step 3 is 1.75, then the z-score for which 92% of the distribution's area lies between minus z and z is -1.75 and 1.75.
Alternatively, you can use a statistical calculator or software to find the z-scores directly by inputting the cumulative probability of 0.04. The calculator will give you the corresponding z-scores.
Remember that the standard normal distribution is symmetric, so the z-scores that divide the area into two equal parts will have the same absolute value but opposite signs.
The area between -1.75 and 1.75 is equal to .91988 or .92
I would use those z-values.
You can get these values from looking at a z-table. There are many different version of z-tables. You might be able to see the tails in it or a cumulative area z-table. Think of 92% being in the middle leaving 4% in each tail or .04,