Question

Find the z-scores that bound the middle 66% of the area under the standard normal curve. Enter the answers in ascending order. Round the answers to two decimal places

Answer 1

The z-scores that bound the middle 66% of the area under the standard normal curve are approximately -0.44 and 0.44.

Step-by-step explanation:

To find the z-scores that bound the middle 66% of the area under the standard normal curve, we start by recognizing that the middle 66% leaves 34% in the tails, splitting into 17% in each tail. From the standard normal (z) distribution, a total of 0.5 plus 0.17 (middle area plus one tail) gives us 0.67. Using the z-table, we find the z-score that corresponds to an area of 0.67 to the left. This score is approximately 0.44. Because the normal curve is symmetric, the z-score that bounds the other side of the middle 66% will be the negative of this value, -0.44. Therefore, the z-scores we're looking for are approximately -0.44 and 0.44, in ascending order.

• Locate the center portion (66%) and tails (17% each) on the standard normal distribution
• Use z-table to find z-scores that correspond to the area to the left
• Account for symmetry in the normal distribution for the second z-score

To conclude, the z-scores that bound the middle 66% of the area under the standard normal curve are approximately -0.44 and 0.44, after rounding to two decimal places.

Answer 2

To find the z-scores that bound the middle 66% of the area under the standard normal curve, we identify the z-scores that leave 17% in the tails. These are approximately -0.95 and 0.95, rounded to two decimal places.

Step-by-step explanation:

The student is asking to find the z-scores that correspond to the middle 66% of the area under the standard normal curve. Because the total area under the standard normal curve equals 1 (representing 100%), the middle 66% implies that we are looking for the z-scores that leave 17% of the area in each of the tails, since 100% - 66% = 34%, and 34% / 2 = 17% for each tail.

Using a z-table, we can find the z-score for the area to the left that corresponds to 0.17. This is the same as the area to the right of the z-score that bounds the upper portion of the middle 66%. The z-table tells us that the z-score that corresponds to an area of 0.83 to the left (which is 1 - 0.17) is approximately 0.95. So, the z-score for the upper side of the middle 66% area is 0.95.

To find the lower side, since the z-distribution is symmetrical, the z-score on the lower side will be -0.95. Therefore, the z-scores that bound the middle 66% of the area under the standard normal curve are -0.95 and 0.95, rounded to two decimal places.