Question

1. the eastern white pine tree is native to the state of michigan. a forester in michigan took a random sample of 50 eastern white pine trees and measured the height of each. the results are shown in the dot - plot. n mean sd min q1 med q3 max 50 64.8 7 53 60 64 68 80 a. what percent of the values are within 1 standard deviation of the mean? 2 standard deviations of the mean? 3 standard deviations of the mean? b. is it reasonable to claim that the distribution of tree heights is approximately normal? explain. 7. which value is the best estimate for the standard deviation of the normal distribution shown in the figure? a) 1 b) 2 c) 5 d) 12 e) 28

Explanation:

Step1: Recall empirical rule for normal distribution

For a normal distribution, approximately 68% of the values are within 1 standard - deviation of the mean, 95% are within 2 standard - deviations of the mean, and 99.7% are within 3 standard - deviations of the mean.

Step2: Analyze normality for tree - height data

For part b, in a normal distribution, the mean, median, and mode are approximately equal. Here, mean = 64.8 and median = 64. Also, in a normal distribution, the inter - quartile range (IQR) is related to the standard deviation. IQR=Q3 - Q1=68 - 60 = 8. For a normal distribution, IQR≈1.35×SD. Given SD = 7, 1.35×7≈9.45 which is close to 8. Also, looking at the dot - plot, it has a somewhat bell - shaped appearance. So it is reasonable to claim the distribution is approximately normal.

Step3: Estimate standard deviation for normal curve

For a normal distribution, about 95% of the data lies within 2 standard deviations of the mean. The range of the data in the normal curve is approximately from 22 to 34. The mean is around 28. The range within 2 standard deviations is 34 - 22 = 12. So 2×SD≈12, then SD≈6. The closest value to 6 among the options is 5.

Answer:

a. Approximately 68% of the values are within 1 standard deviation of the mean, approximately 95% are within 2 standard deviations of the mean, and approximately 99.7% are within 3 standard deviations of the mean.
b. It is reasonable to claim that the distribution of tree heights is approximately normal because the mean and median are close (mean = 64.8, median = 64), the relationship between the inter - quartile range and the standard deviation is consistent with that of a normal distribution, and the dot - plot has a somewhat bell - shaped appearance.

1. C. 5