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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.
5th 10th 25th 50th 75th 90th 95th
69.6 70.9 75.2 81.3 95.4 107.1 116.4
(b)Suppose that the population mean waist size is 85 cm and that the population standard deviation is 14 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.)
(c) Referring back to (b), suppose now that the population mean waist size in 80 cm. Now what is the (approximate) probability that the sample mean will be at least 86.3 cm? (Round your answers to four decimal places.)

Respuesta :

the probability that a random sample of 277 individuals will result in a sample mean waist size of at least 86.3 cm is 0.9982.(c)The probability that a random sample of 277 individuals will result in a sample mean waist size of at least 86.3 cm is 0.9817.

To answer (b), we first need to calculate the z-score for 86.3 cm using the population mean of 85 cm and standard deviation of 14 cm. This gives us a z-score of 0.7. We can then use the normal distribution table to calculate the probability of a sample mean of at least 86.3 cm with a z-score of 0.7, which is 0.9982. To answer (c), we need to calculate the z-score for 86.3 cm using the population mean of 80 cm and standard deviation of 14 cm. This gives us a z-score of 1.5. We can then use the normal distribution table to calculate the probability of a sample mean of at least 86.3 cm with a z-score of 1.5, which is 0.9817.

Part (b):

Z-score = (86.3 - 85) / 14

= 0.7

Probability = 0.9982

Part (c):

Z-score = (86.3 - 80) / 14

= 1.5

Probability = 0.9817

Learn more about probability here

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