Here are some commonly used confidence intervals, with the confidence level in red and the precision in green. What happens though is the interval gets larger, so of course it’s more likely to contain μ. When we add a higher level of confidence, that is going from 95% to 99%, we have more confidence that the population mean is in our interval. The 95% represents our confidence that our interval contains the population mean (the population mean, μ, is the value we are interested in but can never really know). You can also construct 99%, 90% or 12% or 25% confidence intervals. On the other hand, when the ratio of variance between samples is less than 0.05, an unequal variance is assumed and a Welch’s t-test (also for unpaired data sets) is executed.The confidence intervals we use most often are the 95% confidence intervals, however these are not the only ones. For a ratio of variance greater than or equal to 0.05, an equal variance is assumed and a two-sample unpaired t-test is given. Consequently, for unpaired groups, a variance test based on the ratio of the homogeneity of variance between each group is carried out to determine the type of t-test to be performed. Normality assumption is further verified with Shapiro Wilk test calculating a W-statistic. A normal distribution for a Q-Q plot is observed when all the data points lie on the red line. Histograms with embedded density plots, box plots, and normality plots are shown to visualize individual and group differences. A statistical summary of the data comprising of the mean, confidence interval, median, variance, standard deviation, minimum, maximum, and count is provided to quickly communicate the observations in the sample data. A negative t-statistic can be treated as their positive counterpart. For t-statistic less than the t-critical at a p value greater than 0.05 (95% confidence interval), the null hypothesis is accepted. Under the null hypothesis, which states that the means are equal, a t-statistic is calculated that follows a t-distribution with the associated degrees of freedom and a p value is obtained representing the probability that the null hypothesis is true. Both methods assume a continuous data that is randomly selected and normally distributed with equal variances. While the one-sample t-test analyzes the mean of one group against the set average (theoretical mean), the two-sample t-test compares means of two different samples for paired or unpaired data. Statistical t-tests are useful in determining and comparing significant differences between group means and evaluating if those differences are a result of chance.
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