I need help with the ones I got wrong The distribution of the number of hours of

Question

I need help with the ones I got wrong The distribution of the number of hours of sleep people get per night is unimodal and symmetric with a mean of 6 hours and a standard deviation of 1.5 hours. What percent of the people sleep between 6 and 7.5 hours per night? In a group of 800 students, how many would you expect to get more than 9 hours of sleep per night? If James had a z-score of -1.2, how many hour did he sleep? If James had a z-score of -1.2 explain what this value means in context. If Amanda slept 9.2 hours last night, would you consider this unusual? Explain. What is your z-score for how much sleep you got last night? Show work or explain. Mrs. Diaz has two children: a three-year-old boy 43 inches tall and a ten-year-old girl 59 inches tail. Three-year-old boys have a mean height of 38 inches and a standard deviation of 2 inches, and ten-year-old girls have a mean height of 54.5 inches and a standard deviation of 2.5 inches. Assume the distributions of boys' and girls' heights are unimodal and symmetric. Which of Mrs. Diaz's children is taller relative to his or her age and gender? Support your reasoning.

Explanation / Answer

SolutionA:

Normal distribution with mean=6 hours

std dev=1.5 hours

convert the variable to z

z=x-mean/stddev

z=6-6/1.5

=0

z for 7.5 hrs

z=7.5-6/1.5=1

need to find probability

P(6<x<7.5)

=P(0<z<1)

=P(Z=1)-P(Z=0)

Use the standard normal table

=0.8413-0.500

=0.3413

P ( 0<Z<1 )=0.3413

=0.3413*100=34.13%

34.13% of the people sleep between 6 and 7.5 hours

Solutionb:

Since =6 and =1.5 we have:

P ( X>9 )=P ( X>96 )=P ( X/>96/1.5)

Since Z=x/ and 96/1.5=2 we have:

P ( X>9 )=P ( Z>2 )

Step 3: Use the standard normal table to conclude that:

P (Z>2)=0.0228

we have n=sample size=n=400

400*0.0228=9.12

rounding to integer

9 people in 800 sleep more than 9 hours

Solutionc:

z=x-mean/stddev

-1.2=x-6/1.5

-1.2*1.5+6=x

x=4.2 hours

jennes slept 4.2 hours of sleep

Solutiond:

Z-scores simply indicate how many standard deviations away from the mean is a particular score. This is termed "relative standing" as it is a measure of where in the data the score is relative to the mean and "standardized" by the standard deviation. The formula for z is:

z = (x - mean) ÷ standardDeviation

-1.2 means his sleep lies 1.2 standard deviations below mean

it is unsual

Solutione:

x=9.2

z=x-mean/stddev

=9.2-6/1.5

=2.133

z score lies 2.133 standard deviations about mean

hence usual

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