Chapter 6: The Normal Distribution and The Central Limit Theorem
6.8 Using Technology for the Normal Approximation to the Binomial
Learning Objectives
By the end of this section, the student should be able to:
- use technology to solve problems involving the Normal Approximation to the Binomial
This is another section to show technology tools that can be used to help with the statistics concepts taught in this chapter. This section will cover tools to help solve problems involving the Normal Approximation to the Binomial.
Graphing Calculator
Similar to all the other technology sections, the main tool introduced will be the TI-83, 83+, 84, or 84+ Graphing Calculator. The graphing calculator can help to do normal approximation to the binomial problems as seen in 6.5 The Normal Approximation to the Binomial.
Let's look at an example from that section.
Example
Suppose in a local Kindergarten through 12th grade (K - 12) school district, 53 percent of the population favor a charter school for grades K through 5. A simple random sample of 300 is surveyed.
- Find the probability that at least 150 favor a charter school.
- Find the probability that at most 160 favor a charter school.
- Find the probability that more than 155 favor a charter school.
- Find the probability that fewer than 147 favor a charter school.
- Find the probability that exactly 175 favor a charter school.
Let [latex]X =[/latex] the number that favors a charter school for grades K through 5. [latex]X \sim B(n, p)[/latex] where [latex]n = 300[/latex] and [latex]p = 0.53[/latex]. Since [latex]np > 5[/latex] and [latex]nq > 5[/latex], use the normal approximation to the binomial.
Remember, the formulas for the mean and standard deviation are [latex]\mu = np[/latex] and [latex]\sigma =\sqrt{npq}[/latex]. The mean is 159 and the standard deviation is 8.6447. The random variable for the normal distribution is [latex]Y: Y \sim N(159, 8.6447)[/latex].
Remember in Section 6.2, we learned how to use the [latex]\text{normalcdf()}[/latex] function in the graphing calculator:
Using the TI-83+ and TI-84 Calculators for Normal Probabilities
Go into 2nd DISTR.
Press 2: normalcdf.
The syntax for the instructions are as follows: [latex]\text{normalcdf(lower value, upper value, mean, standard deviation)}[/latex]
In some instances:
- the upper number of the area might be [latex]1\text{E}99[/latex] (which equals [latex]10^{99}[/latex]). You get [latex]1\text{E}99 = 10^{99})[/latex] by pressing [latex]1[/latex], the [latex]\text{EE}[/latex] key (a 2nd key) and then [latex]99[/latex]. Or, you can enter [latex]10^{99}[/latex] instead. The number [latex]10^{99}[/latex] is way out in the right tail of the normal curve.
- the lower number of the area might be [latex]-1 \text{E} 99[/latex] (which equals [latex]-10^{99}[/latex]). The number [latex]-10^{99}[/latex] is way out in the left tail of the normal curve.
Use that function for this problem!
Solution
- Include 150 so [latex]P(X \ge 150)[/latex] has normal approximation [latex]P(Y \ge 149.5) = 0.8641[/latex]. [latex]\text{normalcdf}(149.5, 10^{99}, 159, 8.6447) = 0.8641[/latex].
- Include 160 so [latex]P(X \le 160)[/latex] has normal approximation [latex]P(Y \le 160.5) = 0.5689[/latex]. [latex]\text{normalcdf}(0, 160.5, 159, 8.6447) = 0.5689[/latex]
- Exclude 155 so [latex]P(X > 155)[/latex] has normal approximation [latex]P(Y > 155.5) = 0.6572[/latex]. [latex]\text{normalcdf}(155.5, 10^{99}, 159, 8.6447) = 0.6572[/latex]
- Exclude 147 so [latex]P(X \lt 147)[/latex] has normal approximation [latex]P(Y \lt 146.5) = 0.0741[/latex]. [latex]\text{normalcdf}(0, 146.5, 159, 8.6447) = 0.0741[/latex]
- [latex]P(X = 175)[/latex] has normal approximation [latex]P(174.5 \lt Y \lt 175.5) = 0.0083[/latex]. [latex]\text{normalcdf}(174.5, 175.5, 159, 8.6447) = 0.0083[/latex]
Note
Because of calculators and computer software that let you calculate binomial probabilities for large values of n easily, it is not necessary to use the normal approximation to the binomial distribution, provided that you have access to these technology tools. Most school labs have Microsoft Excel, an example of computer software that calculates binomial probabilities. Many students have access to the TI-83 or 84 series calculators, and they easily calculate probabilities for the binomial distribution. If you type in "binomial probability distribution calculation" in an Internet browser, you can find at least one online calculator for the binomial.
For the previous example, the probabilities are calculated using the following binomial distribution: ([latex]n = 300[/latex] and [latex]p = 0.53[/latex]). Compare the binomial and normal distribution answers.
- [latex]P(X \ge 150) : 1 - \text{binomcdf}(300, 0.53, 149) = 0.8641[/latex]
- [latex]P(X \le 160) : \text{binomcdf}(300, 0.53, 160) = 0.5684[/latex]
- [latex]P(X > 155) : 1 - \text{binomcdf}(300, 0.53, 155) = 0.6576[/latex]
- [latex]P(X \lt 147) : \text{binomcdf}(300, 0.53, 146) = 0.0742[/latex]
- [latex]P(X = 175) : \text{binomcdf}(300, 0.53, 175) = 0.0083[/latex]
Helpful Videos for the Graphing Calculator
Below are links to helpful videos for using the graphing calculator for the concepts covered on this page:
Additional Technology Tools
In addition to the graphing calculator, there are some additional technology tools that can be used for the concepts covered on this page. Below are links to helpful videos for those tools:
Section Practice
