Question in statistics and probability help on thermostat.

Question

Readings on a thermometer are normally distributed with a mean of 0C and a standard deviation of 1.00.

What is the probability that a randomly selected

thermometer reads: less than 0.45, greater than 2.11, between 1.05 and 2.20, greater than 2.73, less than -1.27 or greater than 1.27

Answer 1

Given:

Mean = 0

Standard Deviation= 1.00

let X defines the probability wanted,

then X ~ N(0,1)

formula: Z = (X - mean) / standard deviation)

Probability that the thermometer reads:

1) P(X > 0.45)

z = (0.45 - 0) / 1 = 0.54

P(x> 0.45) = P(Z> 0.45) = 0.3264 ( from a normal distribution table)

2) P(X > 2.11)

z = (2.11 - 0 ) / 1 = 2.11

P(x > 2.11)= P(Z > 2.11)= 0.0174 (from a normal distribution table)

3) P(1.05 < X < 2.20)

Z= (1.05-0)/1 = 1.05

Z = ( 2.20- 0)/1 = 2.20

P( 1.05< X < 2.20) = P ( 1.05< Z < 2.20) = 0.133

4) P( X > 2.73) = P (Z > 2.73) ( follow same steps as one and two.

P ( X > 2.73) = 0.0032

5) P ( X < -1.27) = P( Z < - 1.27) ( follow same steps as one and two)

P ( Z < -1.27) = 0.102

6) P ( X > -1.27)= P (Z > -1.27) ( follow same steps as previous one)

P ( X > -1.27)= 0.898

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Answer 2

Given a mean of 0 and a standard deviation of 1, you can use the standardized normal probability table directly, i.e., if x is the thermostat reading and want to find P(x < 0.45), it is the same as P(z < 0.45), which can read from the standard normal probability table or found using a calculator.

P(x > 2.11) = P(z > 2.11) = 1 - P(z < 2.11)

P(1.05 < x < 2.2) = P(1.05 < z < 2.2) = P(z < 2.2) - P(z < 1.05)

P(x > 2.73) = P(z > 2.73) = 1 - P(z < 2.73)

P(x < -1.27) or P(x > 1.27) = 2 * P(x < -1.27) since normal distribution is symmetric = 2 * P(z < -1.27)