Topcis

1. Confidence Intervals: The Basics
   • To estimate an unknown population parameter, choose a statistic that provides a resonable guess. That statistic is the point estimator for the parameter and it's specific value gives a point estimate.
   • A C% confidence interval uses sample data to estimate an unknown population parameter with an indication of how precise the estimate is and of how confident we are that the result is correct.
     • Calculated via $\textrm{point estimate} \pm ME$
     • Otherwise written as $\textrm{statistic} \pm (\textrm{CV})(\textrm{SD of statistic})$
   • To interpret a C% confidence interval, say, "We are C% confident that the interval from _____ to _____ captures the [parameter in context].
   • The margin of error of a condifence interval gets smaller as
     • The confidence level C decreases
     • The sample size n increases
   • The margin of error for a confidence interval includes only chance variation (not other sources like nonresponse and undercoverage).
2. Estimating a Population Proportion
   • The conditions for constructing a confidence interval about a population proportion are:
     • Random: The data were produced by a well-designed random sample or randomized experiment
       • 10%: When sampling without replacement, we check that the population is at least 10 times as large as the sample
     • Large Counts: The sample must be large enough that $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$, otherwise the distribution cannot be assumed to be normal
   • Confidence intervals for a population proportion p are based on the sampling distribution of the sample proportion $\hat{p}$. If the conditions are met, the sampling distribution of $\hat{p}$ is approximately Normal with mean p and SD $\sqrt{p(1-p)/n}$
   • The sample size needed to obtain a confidence interval with approximate margin of error (ME) for a population involves solving for n in the inequality $z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \leq \textrm{ME}$
     • \hat{p} will be the guessed value for the sample proportion (if unknown, 0.5 should be choosen)
3. Estimating a Population Mean
   • Confidence intervals for the mean $\mu$ of a Normal population are based on the sample mean $\bar{x}$ of an SRS. If we know $\sigma$, we can use a z crtical value. Otherwise, which is often the case, we need to use a t critical value
     • This changes the formula $SE_\bar{x}=\frac{\sigma}{\sqrt{n}}$ to $SE_\bar{x}=\frac{s_x}{\sqrt{n}}$
     • degrees of freedom (df) is found by $\textrm{df}=1-n$
   • The conditions for constructing a confidence interval about a population mean are:
     • Random: The data came from a well-designed random sample or experiment
       • 10%: When sampling without replacement, check that the population is at least 10 times as large as the sample.
     • Normal / Large Sample: If the population distribution isn't normal, you must make sure $n \geq 30$. If the sample size is less than 30, check a graph to look for departures from Normality (strong skewness and outliers).
   • The t relys on knowing n (to calculate df), we cannot easily determine a sample size. If possible, look for a similar study with a known $\sigma$ and then solve for n using $z^{*}\frac{\sigma}{\sqrt{n}} \leq \textrm{ME}$

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