Confidence Intervals - Flashcard set - Statistics

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T-Interval for a Population Mean (σ unknown)

To calculate a T-interval, use the formula:

\bar{x} \pm t\frac{s}{\sqrt{n}}

. Interpretation: We are ##% confident that the true population mean falls within this interval.

Log-Normal Distribution Mean Confidence Interval

To calculate the CI for the mean of a log-normal distribution:

\exp\left(\bar{x} \pm z\frac{s}{\sqrt{n}}\right)

Interpretation: We are ##% confident that the true mean of the log-normal distribution falls within this interval.

Paired T-Interval

To calculate a Paired T-Interval, use the formula:

\bar{d} \pm t\frac{s_d}{\sqrt{n}}

. Interpretation: We are ##% confident that the mean difference between the paired data falls within this interval.

Confidence Interval for the Ratio of Two Variances

To calculate this interval, use the formula:

\left(\frac{s_1^2 / s_2^2}{F_{1-\alpha/2, df1, df2}}, \frac{s_1^2 / s_2^2}{F_{\alpha/2, df1, df2}}\right)

Interpretation: We are ##% confident that the ratio of the two population variances falls within this interval.

Confidence Interval for Exponential Distribution Mean

To calculate this interval, use the formula:

\left(\frac{2\sum x_i}{\chi_{\alpha/2}^2}, \frac{2\sum x_i}{\chi_{1-\alpha/2}^2}\right)

Interpretation: We are ##% confident that the true mean of the exponential distribution falls within this interval.

Confidence Interval for a Population Median

For a large sample size, approximate using the normal distribution:

\tilde{m} \pm z\frac{1}{2\sqrt{n}}

Interpretation: We are ##% confident that the true population median falls within this interval.

Confidence Interval for Population Variance

To calculate this interval, use the formula:

\left(\frac{(n-1)s^2}{\chi_{\alpha/2}^2}, \frac{(n-1)s^2}{\chi_{1-\alpha/2}^2}\right)

Interpretation: We are ##% confident that the true population variance falls within this interval.

Z-Interval for a Population Mean (σ known)

To calculate a Z-interval, use the formula:

\bar{x} \pm z\frac{σ}{\sqrt{n}}

. Interpretation: We are ##% confident that the true population mean falls within this interval.

Two-Independent-Samples T-Interval for Difference in Means

To calculate this interval, use the formula:

(\bar{x}_1 - \bar{x}_2) \pm t\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

. Interpretation: We are ##% confident that the difference in population means falls within this interval.

Confidence Interval for Standard Deviation

To calculate this interval, use the formula:

\left(\sqrt{\frac{(n-1)s^2}{\chi_{\alpha/2}^2}}, \sqrt{\frac{(n-1)s^2}{\chi_{1-\alpha/2}^2}}\right)

Interpretation: We are ##% confident that the true population standard deviation falls within this interval.

Confidence Interval for a Population Correlation Coefficient

To calculate this interval, use the Fisher z-transformation:

\tanh\left(\arctanh(r) \pm \frac{z}{\sqrt{n-3}}\right)

Interpretation: We are ##% confident that the true population correlation coefficient falls within this interval.

One-Proportion Z-Interval

To calculate a One-Proportion Z-Interval, use the formula:

\hat{p} \pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

. Interpretation: We are ##% confident that the true population proportion falls within this interval.

Confidence Interval for the Slope of a Regression Line

To calculate this interval, use the formula:

b \pm t\frac{s_e}{\sqrt{SS_{xx}}}

Interpretation: We are ##% confident that the true slope of the population regression line falls within this interval.

Two-Independent-Samples Z-Interval for Difference in Means

To calculate this interval, use the formula:

(\bar{x}_1 - \bar{x}_2) \pm z\sqrt{\frac{σ_1^2}{n_1} + \frac{σ_2^2}{n_2}}

. Interpretation: We are ##% confident that the difference in population means falls within this interval.

One-Proportion Z-Interval for Proportion Difference

To calculate this interval, use the formula:

(\hat{p}_1 - \hat{p}_2) \pm z\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}

. Interpretation: We are ##% confident that the difference in population proportions falls within this interval.

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