Guided Learning Extension
In the Probability Unit, we were able to calculate binomial distribution proportions by adding up the probabilities of each possible event happening. Now that we've learned normal distribution, we are able to approximate proportions without having to do all of that work. Let's look at an example comparing binomial distribution and approximating with normal distribution.
If you are trying to approximate a proportion with a normal distribution, there's one check you have to do before you begin. To see if you can use a normal distribution, multiply the number of possibilities times the probability and multiply the number of possibilities times (1- the probability). Both of these products should be greater than or equal to \(10\).
\(np\ge10\)
\(n(1-p)\ge10\)
Example 1: If a baseball player's batting average is \(0.34\), find the probability that the player will get at most \(40\) hits in his next \(100\) times at bat.
Solving with Binomial Distribution
\(_{100}C_{0}\left(.34\right)^0\left(.66\right)^{100}+_{100}C_{1}\left(.34\right)^1\left(.66\right)^{99}+...+_{100}C_{40}\left(.34\right)^{40}\left(.66\right)^{60}\)
\(=.9137\)
Approximating with Normal Distribution
\(\begin{align}&100\cdot(.34)=34\ \ \ \ \ \ \ 100\cdot(.66)=66\ & \ \ \ \ &\text{1) Check to see you if you can use the normal approximation.}\\
&\mu=np=34\ & \ \ \ \ &\text{2) Find the mean.}\\
&\sigma=\sqrt{np(1-p)}=\sqrt{100(.34)(.66)}=4.74\ & \ \ \ \ &\text{3) Find the standard deviation.}\\
&normalcdf(-\infty, 40, 34, 4.74)=.8974\ & \ \ \ \ &\text{4) Draw a picture representing the situation and calculate the proportion.}\end{align}\)
In the Probability Unit, we were able to calculate binomial distribution proportions by adding up the probabilities of each possible event happening. Now that we've learned normal distribution, we are able to approximate proportions without having to do all of that work. Let's look at an example comparing binomial distribution and approximating with normal distribution.
If you are trying to approximate a proportion with a normal distribution, there's one check you have to do before you begin. To see if you can use a normal distribution, multiply the number of possibilities times the probability and multiply the number of possibilities times (1- the probability). Both of these products should be greater than or equal to \(10\).
\(np\ge10\)
\(n(1-p)\ge10\)
Example 1: If a baseball player's batting average is \(0.34\), find the probability that the player will get at most \(40\) hits in his next \(100\) times at bat.
Solving with Binomial Distribution
\(_{100}C_{0}\left(.34\right)^0\left(.66\right)^{100}+_{100}C_{1}\left(.34\right)^1\left(.66\right)^{99}+...+_{100}C_{40}\left(.34\right)^{40}\left(.66\right)^{60}\)
\(=.9137\)
Approximating with Normal Distribution
\(\begin{align}&100\cdot(.34)=34\ \ \ \ \ \ \ 100\cdot(.66)=66\ & \ \ \ \ &\text{1) Check to see you if you can use the normal approximation.}\\
&\mu=np=34\ & \ \ \ \ &\text{2) Find the mean.}\\
&\sigma=\sqrt{np(1-p)}=\sqrt{100(.34)(.66)}=4.74\ & \ \ \ \ &\text{3) Find the standard deviation.}\\
&normalcdf(-\infty, 40, 34, 4.74)=.8974\ & \ \ \ \ &\text{4) Draw a picture representing the situation and calculate the proportion.}\end{align}\)