Statistical Inference: A Summary of Formulas and Methods



Statistical Inference: A Summary of Formulas and Methods

Written by Professor Jerry Reiter

The table below summarizes the mathematical quantities needed for statistical inference, including standard errors (SE). All confidence intervals are of the form [pic]. The multiplier is derived from either a normal distribution or a t-distribution with some degrees of freedom (abbreviated as “df”).

All hypothesis tests ask the question, how likely are we to observe a value of some test statistic as or more extreme than what was realized in the data, assuming the null hypothesis is true. Test statistics are of the form, [pic]. In most settings involving the difference in two population means or two population percentages, the hypothesized value equals zero, since the null hypothesis is usually that the two population means or two population percentages are equal. The p-value (the chance of observing a test statistic as or more extreme than what was realized, assuming the null hypothesis is true) is determined by looking in the appropriate normal or t-curve table.

|Target of inference|Estimate of target |Standard error for CIs |Test statistic |Lookup table |Comment |

| | | | | | |

|Difference in two |Difference in two |[pic] |[pic]. |Normal |Ask your |

|population |sample percentages | | |curve |instructor which|

|percentages |[pic] | |Or, if the null hypothesis assumes that [pic] | |test statistic |

|[pic] | | | | |is used in your |

| | | |[pic]. | |course. |

| | | | | | |

| | | |where [pic]. | | |

| | | | | | |

| | | |Usually, [pic] in the null hypothesis. | | |

| | | | | | |

|Single population |Sample mean ([pic])|[pic] |[pic] |t-curve with (n-1) df|Use t-curves for|

|mean ([pic]) | |where s is the sample standard deviation of the x|where [pic] is the hypothesized value of the | |averages |

| | |values. |population mean [pic]. | | |

| | | | | | |

|Difference in two |Difference in |[pic] |[pic] |t-curve with |Ask your |

|population means |sample means of two|where [pic] is the sample standard deviation of | |Welch-Satterthwaite |instructor which|

|([pic]) |groups ([pic]) |the x values in the first group, and [pic] is the|Usually, [pic] in the null hypothesis. |df. |degrees of |

| | |sample standard deviation of the x values in the | | |freedom is used |

|Two separate | |second group. | |Some texts use a df |in your course. |

|samples. | | | |equal to the minimum | |

| | | | |sample size minus | |

| | | | |one. | |

| | | | | | |

|Difference in two |Sample mean of |[pic] |[pic] |t-curve with (n-1) |Use matched |

|population means |within-pair |where [pic] is the sample standard deviation of | |df. |pairs only when |

|([pic]) |differences |the within-pair differences, and n is the number |Usually, [pic] in the null hypothesis. | |the data are |

| |([pic]). |of pairs. To get [pic], first compute the | | |collected as |

|Matched pairs. | |difference between the observations in each pair.| | |matched pairs. |

| | |Then, using this single column of differences, | | | |

| | |compute the SD using the usual formula for SDs. | | | |

| | |The [pic]is obtained by averaging the numbers in | | | |

| | |this column. | | | |

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