A sleep time of 15.9 hours per day for a newborn baby is at the percentile of the distribution of sleep times for all newborn babies. Assuming the distribution is normal with a standard deviation of 0.5 hour, approximately what is the meantime, in hours per day, for newborn babies?<\/p>\n\n\n\n
A. 15.1
B. 15.3
C. 16.3
D. 16.5
E. 16.7<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To find the mean<\/strong> sleep time for newborn babies based on the provided information, let’s carefully analyze the problem:<\/p>\n\n\n\n In a normal distribution, any value $X$ can be related to the mean $\\mu$<\/strong> and standard deviation $\\sigma$<\/strong> through the z-score<\/strong> formula:<\/p>\n\n\n\n $$ We\u2019re told that 15.9 hours is at a certain percentile<\/strong> (though the exact percentile isn’t given, it is implied<\/strong> based on the answer choices). Let\u2019s suppose this value (15.9 hours) corresponds to a certain z-score from the standard normal distribution.<\/p>\n\n\n\n Let\u2019s test each of the mean (\u03bc)<\/strong> values in the choices:<\/p>\n\n\n\n $$ Using a z-table: z = 1.6<\/strong> corresponds to the 94.52nd percentile<\/strong><\/p>\n\n\n\n $$ $$ $$ $$ If 15.9 hours is at the 5th percentile<\/strong>, the correct mean would be 16.7<\/strong> (choice E).<\/p>\n\n\n\n If 15.9 hours is at the 11th percentile<\/strong>, the mean would be 16.5<\/strong> (choice D).<\/p>\n\n\n\n The question says: “A sleep time of 15.9 hours\/day is at the __ percentile” \u2014 meaning we are to find the mean assuming<\/strong> this percentile is accurate.<\/p>\n\n\n\n So, if 15.9 is at the 5.48th percentile<\/strong>, that would imply:<\/p>\n\n\n\n This question uses the properties of the normal distribution. By assuming the z-score associated with the given value of 15.9 hours and using the standard deviation, we can work backwards to estimate the mean. When 15.9 hours corresponds to a z-score of -1.6 (which matches the 5.48th percentile), solving the formula yields a mean of 16.7 hours\/day<\/strong>. This method is commonly used in statistics to estimate population parameters from percentile data.<\/p>\n","protected":false},"excerpt":{"rendered":" A sleep time of 15.9 hours per day for a newborn baby is at the percentile of the distribution of sleep times for all newborn babies. Assuming the distribution is normal with a standard deviation of 0.5 hour, approximately what is the meantime, in hours per day, for newborn babies? A. 15.1B. 15.3C. 16.3D. 16.5E. 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\n\n\n\nGiven:<\/strong><\/h3>\n\n\n\n
\n
\n\n\n\nStep 1: Understand the relationship<\/strong><\/h3>\n\n\n\n
z = \\frac{X – \\mu}{\\sigma}
$$<\/p>\n\n\n\n
\n\n\n\nStep 2: Try each answer choice to see which one gives a reasonable z-score<\/strong><\/h3>\n\n\n\n
A. \u03bc = 15.1<\/strong><\/h4>\n\n\n\n
z = \\frac{15.9 – 15.1}{0.5} = \\frac{0.8}{0.5} = 1.6
$$<\/p>\n\n\n\nB. \u03bc = 15.3<\/strong><\/h4>\n\n\n\n
z = \\frac{15.9 – 15.3}{0.5} = 1.2 \\rightarrow 88.49th percentile
$$<\/p>\n\n\n\nC. \u03bc = 16.3<\/strong><\/h4>\n\n\n\n
z = \\frac{15.9 – 16.3}{0.5} = -0.8 \\rightarrow 21.19th percentile
$$<\/p>\n\n\n\nD. \u03bc = 16.5<\/strong><\/h4>\n\n\n\n
z = \\frac{15.9 – 16.5}{0.5} = -1.2 \\rightarrow 11.51st percentile
$$<\/p>\n\n\n\nE. \u03bc = 16.7<\/strong><\/h4>\n\n\n\n
z = \\frac{15.9 – 16.7}{0.5} = -1.6 \\rightarrow 5.48th percentile
$$<\/p>\n\n\n\n
\n\n\n\nStep 3: Match percentiles to the value<\/strong><\/h3>\n\n\n\n
\n
\n\n\n\n\u2705 Correct Answer: E. 16.7<\/strong><\/h3>\n\n\n\n
\n\n\n\nConclusion:<\/strong><\/h3>\n\n\n\n