Question

(9). The National Health Statistics Reports dated Oct. 22, 2008, stated that for a sample size of 277 18-year-old American males, the sample mean waist circumference was 86.3 cm. A somewhat complicated method was used to estimate various population percentiles, resulting in the following values.

5^th   10^th   25^th  50^th 75^th 90^th 95^th

69.6 70.9 75.2 81.3 95.4 107.1 116.4

(a) Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning.

1. Since the mean and median are substantially different, and the difference in the distance between the median and the upper quartile and the distance between the median and the lower quartile is relatively large, it seems plausible that waist size is at least approximately normal.
2. Since the mean and median are nearly identical, and the distance between the median and the upper quartile and the distance between the median and the lower quartile are almost the same, it does not seem plausible that waist size is at least approximately normal.
3. Since the mean and median are nearly identical, and the distance between the median and the upper quartile and the distance between the median and the lower quartile are nearly the same, it seems plausible that waist size is at least approximately normal.
4. Since the mean and median are substantially different, and the distance between the median and the upper quartile and the distance between the median and the lower quartile are nearly the same, it seems plausible that waist size is at least approximately normal.
5. Since the mean and median are substantially different, and the difference in the distance between the median and the upper quartile and the distance between the median and the lower quartile is relatively large, it does not seem plausible that waist size is at least approximately normal.

(b)   Make a conjecture on the shape of the population distribution.

1. The upper percentiles stretch much farther than the lower percentiles. Therefore, we might suspect a right-skewed distribution.
2. The lower percentiles stretch much farther than the upper percentiles. Therefore, we might suspect a right-skewed distribution.     
3. The upper percentiles stretch much farther than the lower percentiles. Therefore, we might suspect a left-skewed distribution.
4. The lower percentiles stretch much farther than the upper percentiles. Therefore, we might suspect a left-skewed distribution.
5. It is plausible that waist size is at least approximately normal.

(C)    Suppose that the population mean waist size is 85 cm and that the population standard deviation is 15 cm. How likely is it that a random sample of 277 individuals will result in a sample mean waist size of at least 86.3 cm? (Round your answers to four decimal places.)

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(d)    Referring back to (C), suppose now that the population mean waist size in 82 cm. Now what is the (approximate) probability that the sample mean will be at least 86.3 cm? (Round your answers to three decimal places.)

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(e)     In light of this calculation, do you think that 82 cm is a reasonable value for μ?

1. No 82 cm is not a reasonable value for μ since if the population mean waist size is 82 cm, there would be almost no chance of observing a sample mean waist size of 86.3 cm (or higher) in a random sample of 277 men.
2. Yes 82 cm is a reasonable value for μ since if the population mean waist size is 82 cm, there would be almost no chance of observing a sample mean waist size of 86.3 cm (or higher) in a random sample of 277 men.    
3. Yes 82 cm is a reasonable value for μ since it is almost the same as 50^th percentile 81.3.
4. Yes 82 cm is a reasonable value for μ since if the population mean waist size is 82 cm, there is a reasonably large chance of observing a sample mean waist size of 86.3 cm (or higher) in a random sample of 277 men.
5. No 82 cm is not a reasonable value for μ since if the population mean waist size is 82 cm, there is a reasonably large chance of observing a sample mean waist size of 86.3 cm (or higher) in a random sample of 277 men.

Answer 1

a)

The mean and median are different and percentiles are not even symmetric about the median.the distribution is not normal and it is right skewed

Since the mean and median are substantially different, and the difference in the distance between the median and the upper quartile and the distance between the median and the lower quartile is relatively large, it does not seem plausible that waist size is at least approximately normal.

b)

The upper percentiles stretch much farther than the lower percentiles. Therefore, we might suspect a right-skewed distribution.

c)

d)

e)

No 82 cm is not a reasonable value for μ since if the population mean waist size is 82 cm, there would be almost no chance of observing a sample mean waist size of 86.3 cm (or higher) in a random sample of 277 men.