Mind on Statistics (6th. Ed) Chapter 11 - Estimating Means with Confidence

Introduction

• Explain the difference between quantitative data collected to create paired differences and data collected as two independent samples
• Describe what the standard error of a statistic represents
• Calculate a confidence interval for one mean of the mean of paired differences
• Identify the appropriate confidence interval formula for the difference in means for independent samples, choosing from the pooled and unpooled method
• Interpret confidence intervals for one mean, the mean of paired differences, and the difference in means for independent samples
• Explain when to use a z-distribution and when to use a t-distribution for computing confidence intervals
• Explain the conditions required for using t confidence intervals for means
• Demonstrate how to create a bootstrap confidence interval for one mean, the mean of paired differences, the differences in two means, and several other parameters, given an appropriate dataset

Confidence Intervals for Means

paired data: means that data have been observed in natural pairs:

• The same measurement taken twice on each person under different conditions
• Similar individuals are paired before giving the treatments in an experiment. Each member of a pair then receives a different treatment. The same response variable is measured for all individuals
• Two different variables are measured for each individual. There is interest in the amount of difference between the two variables.

Confidence Intervals - 1 Population Mean

Conditions for t Confidence Interval for One Mean to be valid:

• The population is bell-shaped and a random sample of any size is measured. For small sampmles, the data should show no extreme skewness and should not contain any outliers.
• A large random sample is measured. $n \geq 30$, should be larger if outliers or extreme skewness present.

\[\bar{x} = t* \cdot \frac{s}{\sqrt{n}}\]

Make sure to interpret confidence interval correctly.

“We are C% confident that the computed interval covers the true mean of the population represented by the sample.”

Prove confidence interval formula works

Confidence Interval - Population Mean of Paired Differences

Same conditions as above.

\[\bar{d}\space \pm t* \cdot \frac{s_d}{\sqrt{n}}\]

Confidence Interval - Difference in 2 Population Means

Same conditions as above.

Two-Sample t-interval

Unpooled (General) Case:

\((\bar{x}_1 - \bar{x}_2)\space \pm t* \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\\ \textrm{df} = \frac{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^2}{\frac{1}{n_1-1}(\frac{s_1^2}{n_1})^2 + \frac{1}{n_2-1}(\frac{s_2^2}{n_2})^2}\) The above df formula is Welch’s approximation and is an approximate multiplier. Can also use a conservative approach by choosing the lesser of $n_1 - 1$ or $n_2-1$ df.

Assuming the two populations have the same variance gives a procedure for which the t* multiplier is mathematically correct.

Pooled Case:

\[\textrm{pooled std. dev } = s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}\\ \textrm{pooled s.e. }(\bar{x}_1 - \bar{x}_2) = \sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}\]

Using Simulation to Calculate Confidence Intervals: Bootstrapping for Means and Other Parameters