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(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 somewhatcomplicated method was used to estimate various populationpercentiles, resulting in the following values.

5^th   10^th  25^th  50^th75^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 leastapproximately normal? Explain your reasoning.

1. Since the mean and median are substantially different, and thedifference in the distance between the median and the upperquartile and the distance between the median and the lower quartileis relatively large, it seems plausible that waist size is at leastapproximately normal.
2. Since the mean and median are nearly identical, and thedistance between the median and the upper quartile and the distancebetween the median and the lower quartile are almost the same, itdoes not seem plausible that waist size is at least approximatelynormal.
3. Since the mean and median are nearly identical, and thedistance between the median and the upper quartile and the distancebetween the median and the lower quartile are nearly the same, itseems plausible that waist size is at least approximatelynormal.
4. Since the mean and median are substantially different, and thedistance between the median and the upper quartile and the distancebetween the median and the lower quartile are nearly the same, itseems plausible that waist size is at least approximatelynormal.
5. Since the mean and median are substantially different, and thedifference in the distance between the median and the upperquartile and the distance between the median and the lower quartileis relatively large, it does not seem plausible that waist size isat least approximately normal.

(b)   Make a conjecture on the shape of the populationdistribution.

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

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

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(d)    Referring back to (C), suppose now thatthe population mean waist size in 82 cm. Now what is the(approximate) probability that the sample mean will be at least86.3 cm? (Round your answers to three decimal places.)

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

1. No 82 cm is not a reasonable value for ? since if thepopulation mean waist size is 82 cm, there would be almost nochance 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 ? since if thepopulation mean waist size is 82 cm, there would be almost nochance 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 ? since it isalmost the same as 50^th percentile 81.3.
4. Yes 82 cm is a reasonable value for ? since if thepopulation mean waist size is 82 cm, there is a reasonably largechance 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 ? since if thepopulation mean waist size is 82 cm, there is a reasonably largechance of observing a sample mean waist size of 86.3 cm (or higher)in a random sample of 277 men.