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Discrete Mathematics

Continuous Probability Distributions

Normal Distribution

         

If \(X\) is a normally distributed variable with mean \( \mu = 11\) and standard deviation \( \sigma = 5,\) then what is the probability \( P(X > 21)?\)

Note: Use the standard normal distribution table below, where \(Z\) has mean \(\mu = 0\) and standard deviation \(\sigma = 1.\)

\[\begin{matrix} z & P(0 \leq Z \leq z) \\ 1 & 0.3413 \\ 1.5 & 0.4332 \\ 2.0 & 0.4772 \\ 2.5 & 0.4938 \end{matrix}\]

If \(X\) is a normally distributed variable with mean \( \mu = 13\) and standard deviation \( \sigma = 4,\) then what is the probability \( P(13<X<23 )?\)

Note: Use the following normal distribution table, where \(Z\) is standardization of \(X\) with \(\mu = 0\) and \(\sigma = 1:\)

\[\begin{matrix} z & P(0 \leq Z \leq z) \\ 1 & 0.3413 \\ 1.5 & 0.4332 \\ 2.0 & 0.4772 \\ 2.5 & 0.4938 \end{matrix}\]

A large group of students took a math test, and their scores obey a normal distribution. If the distribution has mean \(62\) and standard deviation \(10,\) what is the percentage of those students who scored higher than \(72?\)

Note: Use the standard normal distribution table below, where \(Z\) has mean \(\mu = 0\) and standard deviation \(\sigma = 1.\)

\[\begin{matrix} z & P(0 \leq Z \leq z) \\ 1 & 0.3413 \\ 1.5 & 0.4332 \\ 2.0 & 0.4772 \\ 2.5 & 0.4938 \end{matrix}\]

There are a total of 400 students in a secondary school. Their heights obey a normal distribution with mean \( 169 \text{ cm}\) and standard deviation \(4 \text{ cm}.\) What is the approximate number of students whose heights are below \(163 \text{ cm}\) or above \( 171 \text{ cm}?\)

Note: Use the standard normal distribution table below, where \(Z\) has mean \(\mu = 0\) and standard deviation \(\sigma = 1.\)

\[\begin{matrix} z & P(0 \leq Z \leq z) \\ 0.5 & 0.1915 \\ 1 & 0.3413 \\ 1.5 & 0.4332 \\ 2.0 & 0.4772 \\ \end{matrix}\]

If \(X\) is a normally distributed variable with mean \( \mu = 16\) and standard deviation \( \sigma = 4,\) then what is the probability \( P(X<26 )?\)

Note: Use the standard normal distribution table below, where \(Z\) has mean \(\mu = 0\) and standard deviation \(\sigma = 1.\)

\[\begin{matrix} z & P(0 \leq Z \leq z) \\ 1 & 0.3413 \\ 1.5 & 0.4332 \\ 2.0 & 0.4772 \\ 2.5 & 0.4938 \end{matrix}\]

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