B Using normal distribution

The following exercises allow you to check whether you have learned to use the normal distribution (reading from tables or computer).

Exercise B.1 Find the area under the standard normal curve between the following pairs of \(Z\) values:

  1. \(z_1=0\) i \(z_2=2\):

  2. \(z_1=0\) i \(z_2=1\):

  3. \(z_1=0\) i \(z_2=3\):

  4. \(z_1=0\) i \(z_2=0.77\):

Exercise B.2 Find the area under the standard normal curve between the following pairs of \(Z\) values:

  1. \(z_1=-2\) i \(z_2=0\):

  2. \(z_1=-1\) i \(z_2=0\):

  3. \(z_1=-1.77\) i \(z_2=0\):

  4. \(z_1=-0.77\) i \(z_2=0\):

Exercise B.3 Find the following probabilities for the standard normal random variable \(Z\):

  1. \(\mathbb{P}(Z = 1) =\)

  2. \(\mathbb{P}(Z \leq 1) =\)

  3. \(\mathbb{P}(Z < 1) =\)

  4. \(\mathbb{P}(Z > 1) =\)

  5. \(\mathbb{P}(Z \geq 0) =\)

  6. \(\mathbb{P}(-1 \leq Z \leq 1) =\)

  7. \(\mathbb{P}(-2 \leq Z \leq 2) =\)

  8. \(\mathbb{P}(-2.44 \leq Z \leq 0.4) =\)

  9. \(\mathbb{P}(-0.44 \leq Z \leq 1.44) =\)

Exercise B.4 Find the following probabilities for the standard normal random variable \(Z\):

  1. \(\mathbb{P}(Z > 1,44) =\)

  2. \(\mathbb{P}(Z < -1,55) =\)

  3. \(\mathbb{P}(0,66 \leq Z \leq 2,44) =\)

  4. \(\mathbb{P}(-1,96 \leq Z \leq -0,44) =\)

  5. \(\mathbb{P}(-2,5 < Z < 1,5) =\)

  6. \(\mathbb{P}(Z \geq -2,5) =\)

  7. \(\mathbb{P}(Z < 2,5) =\)

Exercise B.5 Find the value of z0 for the variable Z following a standard normal distribution such that:

  1. \(\mathbb{P}(Z \geq z_0) = 0.0401\); \(z_0 =\)

  2. \(\mathbb{P}(-z_0 \leq Z \leq z_0) = 0.95\); \(z_0 =\)

  3. \(\mathbb{P}(-z_0 \leq Z \leq z_0) = 0.90\); \(z_0 =\)

  4. \(\mathbb{P}(-z_0 \leq Z \leq z_0) = 0.8740\); \(z_0 =\)

  5. \(\mathbb{P}(-z_0 \leq Z \leq 0) = 0.2967\); \(z_0 =\)

  6. \(\mathbb{P}(-2 \leq Z \leq z_0) = 0.9710\); \(z_0 =\)

  7. \(\mathbb{P}(Z \geq z_0) = 0.5\); \(z_0 =\)

  8. \(\mathbb{P}(Z \geq z_0) = 0.0057\); \(z_0 =\)

Exercise B.6 Find the value of z0 for the variable Z following a standard normal distribution such that:

  1. \(\mathbb{P}(Z \geq z_0) = 0.05\); \(z_0 =\)

  2. \(\mathbb{P}(Z \geq z_0) = 0.025\); \(z_0 =\)

  3. \(\mathbb{P}(Z \leq z_0) = 0.025\); \(z_0 =\)

  4. \(\mathbb{P}(Z \geq z_0) = 0.10\); \(z_0 =\)

  5. \(\mathbb{P}(Z > z_0) = 0.10\); \(z_0 =\)

Exercise B.7 Provide the standardized Z value (z-score) of a measurement from a normal distribution for the following cases:

  1. 1 standard deviation above the mean:

  2. 1 standard deviation below the mean:

  3. measurement equal to the mean:

  4. 2.5 standard deviation below the mean:

  5. 3 standard deviation above the mean:

Exercise B.8 Assume that the random variable X follows a normal distribution with parameters μ=25 and σ=5. Find the standardized z-score corresponding to each of the following x values:

  1. x = 25:

  2. x = 30:

  3. x = 37.5:

  4. x = 10:

  5. x = 50:

  6. x = 32:

Exercise B.9 Assume that the random variable X follows a normal distribution (µ = 11; σ = 2). Find:

  1. \(\mathbb{P}(10 \leq X \leq 12) =\)

  2. \(\mathbb{P}(6 \leq X \leq 10) =\)

  3. \(\mathbb{P}(13 \leq X \leq 16) =\)

  4. \(\mathbb{P}(7.8 \leq X \leq 12.6) =\)

  5. \(\mathbb{P}(X \geq 13.24) =\)

  6. \(\mathbb{P}(X \geq 7.62) =\)

Exercise B.10 Assume that the random variable X follows a normal distribution with parameters µ = 30 and σ = 8. Find x0, such as:

  1. \(\mathbb{P}(X \geq x_0) = 0.5\); \(x_0 =\)

  2. \(\mathbb{P}(X < x_0) = 0.025\); \(x_0 =\)

  3. \(\mathbb{P}(X > x_0) = 0.10\); \(x_0 =\)

  4. \(\mathbb{P}(X > x_0) = 0.95\); \(x_0 =\)