Find the z-scores for which 28% of the distribution’s area lies between -z and z.
The Correct Answer and Explanation is:
To find the z-scores for which 28% of the distribution’s area lies between -z and z, we need to determine the z-scores corresponding to the cumulative probability that leaves 28% of the area in the middle of the normal distribution. This means that the total area in the tails of the distribution is 100% – 28% = 72%. Since the distribution is symmetric, the area in each tail is half of 72%, which is 36%.
Step-by-step solution:
- Area in the tails: The total area in the tails is 72%. Since the distribution is symmetric, the area in each tail is 36% (i.e., 72% ÷ 2 = 36%).
- Area to the left of z: To find the z-scores, we need to find the cumulative probability corresponding to the area to the left of z. The area between -z and z includes the central 28% of the distribution. Therefore, the cumulative area to the left of z will be: 50%+28%=78%50\% + 28\% = 78\%50%+28%=78% This is because the total area to the left of -z is 50%, and adding the 28% area in the middle gives 78%.
- Find the z-score: Now, we need to find the z-score that corresponds to the cumulative probability of 0.78 (or 78%) in the standard normal distribution. Using a standard normal distribution table or a z-score calculator, we find that a cumulative probability of 0.78 corresponds to a z-score of approximately 0.77.
- Conclusion: The z-scores for which 28% of the distribution’s area lies between -z and z are approximately -0.77 and 0.77.
Explanation:
The standard normal distribution has a mean of 0 and a standard deviation of 1. Z-scores are a measure of how many standard deviations a data point is from the mean. For this problem, we used the cumulative area under the curve to find the z-scores that correspond to the middle 28% of the distribution. The value of 0.77 corresponds to the point where 78% of the distribution lies to the left of it. Thus, the z-scores are symmetric around 0, with the negative z-score being -0.77.
