The results of a survey showed that on average children spen

The results of a survey showed that on average, children spend 5.6 hours at PlayStation per week. Suppose that the standard deviation is 1.7 hours and that the number of hours at PlayStation follows a bell-shaped distribution.

a. Use the empirical rule to calculate the percentage of children who spend between 2.2 and 9 hours at PlayStation per week.
b. What is the z-value for a child who spends 7.5 hours at PlayStation per week?
c. What is the z-value for a child who spends 4.5 hours at PlayStation per week?

Solution

Empirical Rule :

1. At 68 % the rule is,
= (Mean - std) and (Mean + std)

2. At 95% the rule is,
= (Mean - 2 × std) and (Mean + 2 × std)

3. At 97.7% the rule is,
= (Mean - 3 × std) and (Mean + 3 × std)


a.
About 68% of the area under the normal curve is within one standard deviation of the mean. i.e.  
(u ± 1s.d)  
So to the given normal distribution about 68% of the observations lie in between  
= ( 5.6 ± 1.7)  
= ( 5.6 - 1.7, 5.6 + 1.7 ) = ( 3.9, 7.3)
  
About 95% of the area under the normal curve is within two standard deviations of the mean. i.e.  
(u ± 2*s.d)  
So to the given normal distribution about 68% of the observations lie in between  
=( 5.6 ± 2*1.7)  
= ( 2.2, 9)
  
b.
Z = X- u / sd
Z = (7.5-5.6)/1.7
= 1.9/1.7= 1.1176

c.
Z = (4.5-5.6)/1.7
= -1.1/1.7= -0.6471