Discrete Mathematics

Continuous Probability Distributions

Normal Distribution

         

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

Note: Use the standard normal distribution table below, where ZZ has mean μ=0\mu = 0 and standard deviation σ=1.\sigma = 1.

zP(0Zz)10.34131.50.43322.00.47722.50.4938\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 XX is a normally distributed variable with mean μ=13 \mu = 13 and standard deviation σ=4, \sigma = 4, then what is the probability P(13<X<23)? P(13<X<23 )?

Note: Use the following normal distribution table, where ZZ is standardization of XX with μ=0\mu = 0 and σ=1:\sigma = 1:

zP(0Zz)10.34131.50.43322.00.47722.50.4938\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 6262 and standard deviation 10,10, what is the percentage of those students who scored higher than 72?72?

Note: Use the standard normal distribution table below, where ZZ has mean μ=0\mu = 0 and standard deviation σ=1.\sigma = 1.

zP(0Zz)10.34131.50.43322.00.47722.50.4938\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 cm 169 \text{ cm} and standard deviation 4 cm.4 \text{ cm}. What is the approximate number of students whose heights are below 163 cm163 \text{ cm} or above 171 cm? 171 \text{ cm}?

Note: Use the standard normal distribution table below, where ZZ has mean μ=0\mu = 0 and standard deviation σ=1.\sigma = 1.

zP(0Zz)0.50.191510.34131.50.43322.00.4772\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 XX is a normally distributed variable with mean μ=16 \mu = 16 and standard deviation σ=4, \sigma = 4, then what is the probability P(X<26)? P(X<26 )?

Note: Use the standard normal distribution table below, where ZZ has mean μ=0\mu = 0 and standard deviation σ=1.\sigma = 1.

zP(0Zz)10.34131.50.43322.00.47722.50.4938\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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