70 lines
4.3 KiB
Markdown
70 lines
4.3 KiB
Markdown
(Ch 19, stat 1040)
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| Term | Definition |
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| ---- | ---- |
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| Qualitative | A descriptive value (red, blue, high, low) |
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| Quantitative | A numerical value (7, 8, 9) |
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| Population | The entire set of existing units that investigators wish to study |
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| Sample | A portion or subset of the population |
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| Parameter | A number that describes a characteristic of an entire *population* (*10%* of US senators voted for something) |
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| Statistic | A number that describes a *sample* characteristic (*71%* of Americans feel that ...) |
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> A global consumer survey reported that 6% of US taxpayers used or owned cryptocurrency in 2020. The US government is interested in knowing if this percentage has increased. The University of Chicago surveys 1,004 taxpayers and finds that 13% have used or owned crypto in the past year (2021)
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In the above example:
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- The *population* was *US taxpayers*
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- The *parameter* was *6%*
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- The *sample* was *1004 taxpayers*
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- The *statistic* was *13%*
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An ideal sample will represent the whole population.
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## Sampling
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| Sample Type | Description |
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| ---- | ---- |
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| Simple random | Advantages:<br>- Procedure is impartial<br>- Law of Averages<br>Disadvantages<br>- Not always possible<br>- Can be very expensive |
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| Quota Sampling | Attempts to get certain proportions based on key characteristics. Quota sampling doesn't guarantee that the selection is an accurate representation. |
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| Cluster Sampling | Divide population into subgroups, randomly select a subgroup, and sample all of the subjects in that group |
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| Convenience | Sampling done near to the researcher because it's easier. |
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## Simple Random Samples
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## Bias
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| Bias Type | Description |
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| ---- | ---- |
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| Selection | When the procedure that selects the sample is biased |
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| Non-Response | Those that don't respond to a survey may have different characteristics than those that do respond |
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| Response | When the question is worded in a leading way to elicit a certain response. |
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| Volunteer response | Self selecting, individuals volunteer to answer |
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| Measurement | Interviewing method influences the response, uses loaded words or ambiguities. |
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## Percentages
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(Ch 20, stat 1040)
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Throughout this chapter, percentages are often represented by referencing a box model of 1s and 0s, where 1s are datapoints that *are* counted, and 0s are datapoints that are not counted.
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The expected value for a sample percentage equals the population percentage. The standard error for that percentage = `(SE_sum/sample_size) * 100%`.
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To determine by how much the standard error is affected, if $n$ is the proportion that the population changed by, the standard error will change by $\frac{1}{\sqrt{n}}$. For example, if a population changed from 20 to 40, that change was by a proportion of 2, and so you would say the standard error decreased by $\frac{1}{\sqrt{2}}$.
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Accuracy in statistics refers to how small the standard error is. A smaller standard error means your data is more accurate. As the sample size increases, the percentage standard error decreases.
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You can use the below equation to find the percentage standard error of a box model that has ones and zeros. the % of ones and zeros should be represented as a proportion (EG: `60% = 0.6`).
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$$ SE_\% = \sqrt{\frac{(\%\space of\space 1s)(\%\space of\space 0s)}{num_{draws}}} $$
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If asked if an observed % is reasonable, you can calculate the z score, and if the z score is more than 2-3 standard deviations away.
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## Sampling Distributions
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(Ch 23, stat 1040)
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Take a sample, find the average, plot it and repeat. After many many samples, the *observed* probability histogram for sample averages looks like the *predicted* probability histogram.
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As with $SE_\%$, as the sample size increase, the standard error decreases.
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The central limit theorem still applies here, so the probability histogram for the average of the draws *follows the normal curve* with a large number of draws, even if the contents of the box do not.
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To calculate the $SE_{ave}$, use the below equation:
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$$ \frac{SD_{box}}{\sqrt{num_{draws}}} $$
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| Term | Definition |
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| ---- | ---- |
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| $EV_{ave}$ | The expected value for the average of the population |
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| $SE_{ave}$ | The standard error |
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## Confidence Interval
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95% confidence means that 95% of the time the interval constructed will capture the parameter, and 95% of the time it will not. |