Question

Fill in the blanks. The depth of the snow in my yard is normally distributed, with a mean of 2.5 inches and a standard deviation of 0.25 inches. What value is one standard deviation below the mean? inches What value is one standard deviation above the mean? inches What is the probability that a randomly chosen location will have a snow depth between 2.25 and 2.75 inches? percent

Answer 1

1 standard deviation below the mean = mean - 1 * SD

= 2.5 - 1 * 0.25

= 2.25

1 standard deviation above the mean = mean + 1 * SD

= 2.5 + 1 * 0.25

= 2.75

P ( 2.25 < X < 2.75 ) = ?
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 2.25 - 2.5 ) / 0.25
Z = -1
Z = ( 2.75 - 2.5 ) / 0.25
Z = 1
P ( -1 < Z < 1 ) =  P ( 2.25 < X < 2.75 ) = P ( Z < 1 ) - P ( Z < -1 )
P ( 2.25 < X < 2.75 ) = 0.8413 - 0.1587
P ( 2.25 < X < 2.75 ) = 0.6827

= 68.27%