Question

recall information from the previous task. one study of 91 bald eagle eggs found that the eggs had a mean length of 73.6 millimeters and standard deviation of 2.9 millimeters. one bald eagle egg was 74.9 millimeters long. the z - score for this egg was 0.45. which statements correctly interpret the z - score? choose two correct answers. the length of this egg was 0.45 standard deviations above the mean length. the egg was 0.45 millimeters longer than the mean. this egg was longer than 45 percent of the eggs in the study. this egg was longer than the mean by 0.45 standard deviations, or 1.3 millimeters. the length of this egg was 0.45 times the mean.

Explanation:

Step1: Define z-score meaning

A z-score $z$ means the data point is $z$ standard deviations from the mean. For $z=0.45$, it is 0.45 standard deviations above the mean.

Step2: Calculate deviation from mean

Deviation = $z \times \text{standard deviation} = 0.45 \times 2.9 = 1.305 \approx 1.3$ millimeters. So the egg is 1.3 mm longer than the mean, which matches the statement about being 0.45 standard deviations above the mean, and the statement about being longer than the mean by 0.45 standard deviations (1.3 mm).

Step3: Evaluate other statements

• 0.45 mm longer than mean: $74.9 - 73.6 = 1.3$ mm, not 0.45 mm. Incorrect.
• 45 percent of eggs shorter: Z-score does not directly give this percentage without a z-table, and this misinterprets the z-score value. Incorrect.
• 0.45 times the mean: $0.45 \times 73.6 = 33.12

eq 74.9$. Incorrect.

Answer:

• The length of this egg was 0.45 standard deviations above the mean length.
• This egg was longer than the mean by 0.45 standard deviations, or 1.3 millimeters.