Question

For of a bell-shaped data with mean of 110 and standard deviation of 11, approximately

a) 0.15% of the values lies below: _________

b) 68% of the middle values lies between:______and______

c) 2.5% of the values lies above: ________

d) 0.15% of the values lies above:_________

Answer 1

Solution:-

Given that,

mean = = 110

standard deviation = = 11

a) Using standard normal table,

P(Z < z) = 0.15%

= P(Z < z) = 0.0015  

= P(Z < -2.97) = 0.0015

z = -2.97

Using z-score formula,

x = z * +

x = -2.97 * 11 + 110

x = 77.33

0.15% of the values lies below: 77.33

b) Using standard normal table,

P( -z < Z < z) = 68%

= P(Z < z) - P(Z <-z ) = 0.68

= 2P(Z < z) - 1 = 0.68

= 2P(Z < z) = 1 + 0.68

= P(Z < z) = 1.68 / 2

= P(Z < z) = 0.84

= P(Z < 0.99) = 0.84

= z  ± 0.99

Using z-score formula,

x = z * +

x = -0.99 * 11 + 110

x = 99.11

Using z-score formula,

x = z * +

x = 0.99 * 11 + 110

x = 120.89

68% of the middle values lies between: 99.11 and 120.89

c) Using standard normal table,

P(Z > z) = 2.5%

= 1 - P(Z < z) = 0.025  

= P(Z < z) = 1 - 0.025

= P(Z < z ) = 0.975

= P(Z < 1.96 ) = 0.975  

z = 1.96

Using z-score formula,

x = z * +

x = 1.96 * 11 + 110

x = 131.56

2.5% of the values lies above: 131.56

d) Using standard normal table,

P(Z > z) = 0.15%

= 1 - P(Z < z) = 0.0015  

= P(Z < z) = 1 - 0.0015

= P(Z < z ) = 0.9985

= P(Z < 2.97 ) = 0.9985  

z = 2.97

Using z-score formula,

x = z * +

x = 2.97 * 11 + 110

x = 142.67

0.15% of the values lies above: 142.67