Chapter 6: The Normal Distribution and The Central Limit Theorem
Chapter 6 Review
Chapter Review
6.1Summary
A z-score is a standardized value. Its distribution is the standard normal, [latex]Z \sim N(0, 1)[/latex]. The mean of the z-scores is zero and the standard deviation is one. If z is the z-score for a value x from the normal distribution [latex]N(\mu, \sigma)[/latex] then z tells you how many standard deviations x is above (greater than) or below (less than) [latex]\mu[/latex].
Formula Review
[latex]X \sim N(\mu, \sigma)[/latex]
[latex]\mu = \text{the mean; } \sigma = \text{the standard deviation}[/latex]
z = a standardized value (z-score)
mean = 0; standard deviation = 1
To find the observed value, x, when the z-scores is known: [latex]x= \mu + (z) \sigma[/latex]
z-score: [latex]z= \frac{x\text{ – }\mu }{\sigma }[/latex]
Z = the random variable for z-scores, [latex]Z \sim N(0, 1)[/latex]
Section 6.2
Summary
The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean [latex]\mu[/latex] and the standard deviation [latex]\sigma[/latex]. A special normal distribution, called the standard normal distribution is the distribution of z-scores. Its mean is zero, and its standard deviation is one.
Formula Review
Normal Distribution: [latex]X \sim N(\mu, \sigma)[/latex] where [latex]\mu[/latex] is the mean and [latex]\sigma[/latex] is the standard deviation.
Standard Normal Distribution: [latex]Z \sim N(0, 1)[/latex].
Calculator function for probability: [latex]\text{normalcdf} (\text{lower }x \text{value of the area, upper }x \text{value of the area, mean, standard deviation})[/latex]
Calculator function for the kth percentile: [latex]k = \text{invNorm} (\text{area to the left of }k \text{, mean, standard deviation)}[/latex]
Section 6.3
Summary
In a population whose distribution may be known or unknown, if the size ([latex]n[/latex]) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size ([latex]n[/latex]).
Formula Review
The Central Limit Theorem for Sample Means: [latex]\overline{X} \sim N \left({\mu }_{x}, \frac{\sigma x}{\sqrt{n}}\right)[/latex]
The Mean [latex]\overline{X}[/latex]: [latex]\mu_x[/latex]
Central Limit Theorem for Sample Means z-score and standard error of the mean: [latex]z=\frac{\overline{x}-{\mu }_{x}}{\left(\frac{{\sigma }_{x}}{\sqrt{n}}\right)}[/latex]
Standard Error of the Mean (Standard Deviation ([latex]\overline{X}[/latex])): [latex]\frac{{\sigma }_{x}}{\sqrt{n}}[/latex]
Section 6.4
Summary
The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution even if the original population is not normally distributed. Additionally, if the original population has a mean of [latex]\mu_x[/latex] and a standard deviation of [latex]\sigma_x[/latex], the mean of the sums is [latex]n \mu_x[/latex] and the standard deviation is [latex]\left(\sqrt{n}\right)(\sigma_x)[/latex] where [latex]n[/latex] is the sample size.
Formula Review
The Central Limit Theorem for Sums: [latex]\Sigma{X} \sim N[(n)(\mu_x), (\sqrt{n})(\sigma_x)][/latex]
Mean for Sums ([latex]\Sigma{X}[/latex]): [latex](n)(\mu_x)[/latex]
The Central Limit Theorem for Sums z-score and standard deviation for sums: [latex]z\text{ (for the sample mean)} = \frac{\Sigma x–\left(n\right)\left({\mu }_{X}\right)}{\left(\sqrt{n}\right)\left({\sigma }_{X}\right)}[/latex]
Standard deviation for Sums ([latex]\Sigma{X}[/latex]): [latex]\left(\sqrt{n}\right)(\sigma_x)[/latex]
Section 6.5
Summary
The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample mean [latex]\overline{x}[/latex] gets to [latex]\mu[/latex].