The following figure shows the normal distribution with the proportion of the ar

Question

The following figure shows the normal distribution with the proportion of the area under the normal curve contained within one, two, and three standard deviations of the mean. The last proportion on each side, 0.13%, depicts the remaining area under the curve. Specifically, 0.13% of the area under the standard normal distribution is located above z-score values greater than the mean () plus three standard deviations (+30). Also, because the normal distribution is symmetrical, 0.13% of the area under the standard normal distribution is located below z-score values less than the mean () minus three standard deviations (-30). 34.13% 34.13% 13.59% 13.59% 2.15% 2.15% 0.13% 0.13% Use the figure to help you answer the following questions The National Assessment of Educational Progress (NAEP) is a nationwide assessment of students' proficiency in nine subjects: mathematics, reading, writing, science, the arts, civics, economics, geography, and U.S. history. The main NAEP assessments are conducted annually on samples of students from grades 4, 8, and 12.

Explanation / Answer

Here x follows normal with mean 264 and standard deviation is 34

For 196, z score is

z=(196-264)/34=-2

For 332, z score is

z=(332-264)/34=2

Hence P(-2<z<2)=P(0<z<2)-P(0<z<-2)=0.4772-(-0.4772)=0.9544

Hence 95.44% of male score between 196 and 332

For value 366, z score is

z=(366-264)/34=3

So P(z<3)=0.9987

Hence 99.87% of males are below 366

Now we need to find x such that P(X>x)=0.8413

z value is P(z>-1)=0.8413

So z=(x-mean)/sd=-1

So x=-1*sd+mean=-1*34+264=230

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