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Continuous Probability Distributions

How much variation should you have in your blood pressure? How likely is that stock price to double by the end of the year? Use continuous probability distributions to find out!

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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