Chapter 18 - Sampling Distribution
| Tuesday, 10 January 2023 | |
| 2-minute read | |
| 346 words | |
Sampling distributions can be most useful with a normal curve. In order to make this assumption, certain conditions need to be met.
Randomization
- the sampling method must be known to be random and guarantee normality
Independence of selections
- Independence must be stated or implied
- OR no more than 10% of the population must be sampled
Large enough sample condition
- For proportions: there must be at least 10 succcesses $np$ and at least 10 failures $n(1 - p)$ which keeps us away from heavily skewed situations
For means: central limit theorem
- When the sample size is at least 30, the sampling distribution will be approximately normal
Proportions
This only applies to proportions.
The mean proportion of a sampling distribution will be $p$, the parameter (the unknown proportion of the whole population). The best estimate for this center would be represented with $\hat{p}$.
Standard Deviation or Standard Error
\[ SE = \sqrt{\frac{pq}{n}} \]
where $p$ is the proportion of successes, $q$ is the proportion of failures $1 - p$, and $n$ is the sample size.
Means
This only applies to means.
The mean of a sampling distribution will be $\mu$, the parameter (the unknown mean of the whole population). The best estimate for this would be represented with $\bar{x}$
Standard Deviation
\[ SE = \frac{\sigma}{\sqrt{n}} \]
Example
Assessment records indicate that the value of homes in a small city is skewed right, with a mean of $140k and standard deviation of $60k. To check the accuracy of the assessment data, officials plan to conduct a detailed appraisal of 100 homes selected at random. Using the 68-95-99.7 Rule, draw and label an appropriate sampling model for the mean value of the homes selected.
\begin{array}{} \mu = 14000 \\ \sigma = 60000 \\ n = 100 \end{array}
- random
- Independent or less than 10%
- sample size at least 100
\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{60000}{10} = 6000 \]
\begin{array}{l | l} \mu - 2 \textrm{SE} & 128000 \\ \mu - \textrm{SE} & 134000 \\ \mu & 140000 \\ \mu + \textrm{SE} & 146000 \\ \mu + 2 \textrm{SE} & 152000 \end{array}