Question

An important characteristic for the production of widgets is length. The process is known to yield an average length equal to 2.5 and a standard deviation of 1.2. Assume that a random length measurement comes from a normal distribution.

- What percent of the data falls above 2.5? Justify Your Answer

- Give the median for this distribution and justify your answer

- What percent of the data falls between 1.3 and 2.7? Justify your answer

Answer 1

Mean length = 2.5

standard deviation = 1.2

X ~ N(2.5,1.44)

What percent of the data falls above 2.5?

P[ X > 2.5 ] = P[ ( X - mean(X))/sd(X) > ( 2.5 - mean(X))/sd(X) ]

P[ X > 2.5 ] = P[ ( X - 2.5)/1.2 > ( 2.5 - 2.5)/1.2 ]

P[ X > 2.5 ] = P[ Z > 0]

P[ X > 2.5 ] = 0.5

P[ X > 2.5 ] = 50%

Give the median for this distribution and justify your answer

For normal distribution mean = median = 2.5

What percent of the data falls between 1.3 and 2.7?

P[ 1.3 < X < 2.7 ] = P[ ( 1.3 - mean(X))/sd(X) < ( X - mean(X))/sd(X) > ( 2.7 - mean(X))/sd(X) ].

P[ 1.3 < X < 2.7 ] = P[ ( 1.3 - 2.5)/1.2 < ( X - 2.5)/1.2 > ( 2.7 - 2.5)/1.2 ]

P[ 1.3 < X < 2.7 ] = P[ -1 < Z < 0.17 ]

P[ 1.3 < X < 2.7 ] = P[ Z < 0.17 ] - P[ Z < -1 ]

P[ 1.3 < X < 2.7 ] = 0.5675 - 0.1587

P[ 1.3 < X < 2.7 ] = 0.4088