The notation for the chi-square distribution is: χ∼χ2dfχ∼χdf2 where df = degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use df = n – 1. The degrees of freedom for the three major uses are each…
Figure 11.1 The chi-square distribution can be used to find relationships between two things, like grocery prices at different stores. (credit: modification of work “The first few times I used my Asda ‘bag for life’ I left the receipts in the bottom…
Comparing Two Independent Population Means Two population means from independent samples where the population standard deviations are not known Cohen’s Standards for Small, Medium, and Large Effect Sizes Cohen’s d is a measure of “effect size” based on the differences between two…
In most cases of economic or business data we have little or no control over the process of how the data are gathered. In this sense the data are not the result of a planned controlled experiment. In some cases,…
Even though this situation is not likely (knowing the population standard deviations is very unlikely), the following example illustrates hypothesis testing for independent means with known population standard deviations. The sampling distribution for the difference between the means is normal…
Proportions have their base in the binomial probability distribution. As a probability distribution, we know the mean and standard deviation of the distribution. As a binary or categorical sample data set, we lose this knowledge from when we knew the…
Typically we can never expect to know any of the population parameters, mean, proportion, or standard deviation. When testing hypotheses concerning differences in means we are faced with the difficulty of two unknown variances that play a critical role in…
Cohen’s d is a measure of “effect size” based on the differences between two means. Cohen’s d, named for United States statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated…
The comparison of two independent population means is very common and provides a way to test the hypothesis that the two groups differ from each other. Is the night shift less productive than the day shift, are the rates of…
Figure 10.1 If you want to test a claim that involves two groups (the types of breakfasts eaten east and west of the Mississippi River) you can use a slightly different technique when conducting a hypothesis test. (credit: modification of work “scrambled…