Normal distribution: exercises

The graphical representation of a normal distribution is a bell-shaped curve that is symmetrical about the mean. Thus, half of the values ​​of the random variable X are equal to or greater than the mean and half are equal to or less than the mean. The curve covers 100% of the population. All possible values ​​that the random variable can assume lie under the curve. A normal distribution is defined by two parameters: the mean, denoted by µ and the standard deviation, denoted by σ.

Graphical representation of the normal distribution

EXERCISES

1.   On a certain road, the speed limit is 40 km/h with a tolerance of 7 km/h. The speed at which the driver travels on this road varies, with an average speed μ=40km/h and a standard deviation σ = 4 km/h. What is the probability that the driver will exceed 47 km/h and get a ticket? To determine this, we must standardize the variable (µ = 0, σ =1. Then we can consult a standardized normal distribution table, typically located at the end of statistical textbooks. Calculate:

Tip: You can utilize software to determine probabilities..

2. In regular operation conditions, radar readings is a random variable normally distributed with mean μ = 25 mph and standard deviation σ = 3 mph. To test the calibration of the radar, a test car traveling at 25 mph is used. Assuming the radar is correctly calibrated, i.e., with μ = 25 mph and σ = 3 mph, what is the probability that the radar detects the test car's speed to be: a) 28 mph or higher? b) 27½ mph or higher? c) At what speed should the radar record for the probability of a value exceeding this speed to be 5%?

    a) Probability of detecting 28 mph or more.

                      b) Probability of detecting 27½ mph or more.

 c) To have a 5% probability of the radar measuring a higher speed.

                                                             Remember that

More control means fewer mistakes. You need to keep this in mind when you want more quality. Compare the probabilities obtained in examples 2b, 3 and 4.

3.  A calibration test will be carried out on 4 radars that operate together. A test car with a speed set at 25 mph will be used.  Under optimal conditions, the speed of each radar is a random variable following a normal distribution with mean µ=25 mph and standard deviation σ=3mph. If the radars are calibrated, what is the probability that the collected measurements will give an average speed of 27½ mph or higher?

4. The idea of ​​using 4 radars is good; however, having 9 is advantageous. Increased control (in this case, through a larger quantity of radars) decreases the risk of error. There are 9 calibrated radars in operation simultaneously. After calibration, the speed of each radar unit is a normally distributed random variable with mean µ = 25 mph and standard deviation σ = 3 mph. Using a test car traveling at a constant speed of 25 mph, what is the probability that the collected measurements will yield an average speed 27½ mph or higher?

OPERATIONS OF RANDOM VARIABLES

If X and Y are independent random variables with means µ₁ and µ₂ and variances σ₁² and σ₂² respectively, then:

Z = X + Y is a random variable with mean µ₁ + µ₂ and variance σ₁² + σ₂².

W = X – Y is a random variable with mean µ₁ - µ₂ and variance σ₁² + σ₂².

Var (a Z) = a² Var (Z)

σ (a Z) = a σ (Z). 

                                             EXERCISE

5. If we consider the subway arrival time as a random variable with mean µ₁ = 8h10m and standard deviation σ₁ = 40s, and your arrival at the station as a random variable with mean µ₂ = 8h08m and standard deviation σ₂ =30s, how likely is it that you will miss the train? Let S represent the subway arrival time and Y represent your arrival time at the station. Therefore, W = S - Y, your waiting time, is a random variable with a normal distribution, having an average of 8:10 – 8:08 = 2 m = 120 s, variance of 402 + 302 =1600 + 900 = 2500 s, and a standard deviation of 50. Missing the train would occur if W ≤ 0. To calculate the probability of W ≤ 0:

Decision rule

Set critical values ​​ to reject specific results. This is crucial for quality control.

                                                EXERCISE

6. In the canning industry, the pH (the acidity) of the product in each can is a random variable that is normally distributed with mean 7 and standard deviation of 0.5. If the pH of the product falls below 6.0 or exceeds 8.0, the can is rejected. What is the probability of this happening?