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Find the -scores for which of the distribution's area lies between and .Click to view bace I I the click to view page of the table.The -scores are .(Use a comma to separate answers as needed. Round to two decimal places as needed.)
Full solution
Q. Find the -scores for which of the distribution's area lies between and .Click to view bace I I the click to view page of the table.The -scores are .(Use a comma to separate answers as needed. Round to two decimal places as needed.)
- Understand the problem: Understand the problem.We need to find two z-scores such that the area between them under the standard normal distribution curve is . Since the normal distribution is symmetric, the area from to will be split equally on both sides of the mean (which is for a standard normal distribution). This means that of the area lies to the left of and to the right of .
- Use the standard normal distribution table: Use the standard normal distribution table.To find the z-scores, we will use the standard normal distribution table, which gives us the area to the left of a given z-score. Since we want (or ) of the area to the left of , we need to find the z-score that corresponds to an area of to the left of it.
- Look up the z-score: Look up the z-score corresponding to an area of . Using the standard normal distribution table, we find that the z-score that leaves an area of to its left is approximately . This is the z-score for .
- Find the positive z-score: Find the positive z-score.Since the normal distribution is symmetric, the positive z-score that leaves an area of to its left () will have the same absolute value as . Therefore, is also approximately .
- Verify the solution: Verify the solution.We have found that the z-scores are approximately and . To verify, we can check that the area between these z-scores is . The area to the left of is , and the area to the left of is . The difference between these areas is , which is . This confirms that our z-scores are correct.
