The results of a national survey showed that on average, adults sleep 6.7 hours per night. Suppose that the standard deviation is 1.8 hours. (a) Use Chebyshev's theorem to calculate the minimum percentage of individuals who sleep between 3.1 and 10.3 hours. % (b) Use Chebyshev's theorem to calculate the minimum percentage of individuals who sleep between 2.2 and 11.2 hours. % (c) Assume that the number of hours of sleep follows a bell-shaped distribution. Use the empirical rule to calculate the percentage of individuals who sleep between 3.1 and 10.3 hours per day. % How does this result compare to the value that you obtained using Chebyshev's theorem in part (a)

a) 75%

b) 84%

c) 95%                                                    

Step-by-step explanation:

We are given the following in the question:

Mean, μ = 6.7 hours

Standard Deviation, σ = 1.8 hours

Chebyshev's Theorem:

Empirical Formula:

a) minimum percentage of individuals who sleep between 3.1 and 10.3 hours

Minimum percentage:

Thus, minimum 75% of individuals who sleep between 3.1 and 10.3 hours.

b) minimum percentage of individuals who sleep between 2.2 and 11.2 hours.

Minimum percentage:

Thus, minimum 84% of individuals who sleep between 2.2 and 11.2 hours.

c) percentage of individuals who sleep between 3.1 and 10.3 hours per day

According to Empirical rule about 95% of the data lies within 2 standard deviations from the mean.

Thus, 95% of individuals who sleep between 3.1 and 10.3 hours.

d) Comparison with Chebyshev's theorem

This is greater than the results obtained from the Chebyshev's theorem in part (a)

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

step-by-step explanation: