8.4 - Lesson 8 Summary

Objectives

Upon successful completion of this lesson, you should be able to:

• Identify situations in which the z or t distributions may be used to approximate a sampling distribution
• Construct a confidence interval to estimate a population proportion, mean, or difference in paired means by hand given summary data
• Construct a confidence interval to estimate a population proportion, mean, or difference in paired means using Minitab Express given summary or raw data
• Determine the necessary minimum sample size to construct a confidence interval for a single proportion or single mean with a given level of confidence and margin of error
• Conduct a hypothesis test using the appropriate common distribution for a single proportion, single mean, and paired means by hand given summary data
• Conduct a hypothesis test using the appropriate common distribution for a single proportion, single mean, and paired means using Minitab Express given summary or raw data

In this lesson you learned how to construct confidence intervals and conduct hypothesis tests for one proportion, one mean, and paired means using the $$z$$ and $$t$$ distributions. These procedures are sometimes referred to as "traditional" methods, in contrast to the bootstrapping and randomization procedures that we learned earlier in the course which are "simulation-based inference" methods.

The table below summarizes the procedures covered in this lesson. Note the similarities between all of the confidence interval formulas and all of the test statistic formulas. These all follow the same general format.

Procedure Assumptions Standard Error Confidence Interval Test Statistic
One Proportion $$z$$

Confidence interval: At least 10 successes and 10 failures in the sample

Hypothesis test: $$n p_0 \ge 10$$ and $$n (1-p_0) \ge 10$$

$$\sqrt{\frac{p(1-p)}{n}}$$ $$\widehat{p} \pm z^{*} \left ( \sqrt{\frac{\hat{p} (1-\hat{p})}{n}} \right)$$ $$z=\frac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}$$
One Mean $$t$$

Sample size is at least 30 OR population is normally distributed

Population standard deviation is unknown*

$$\frac{\sigma}{\sqrt n}$$ $$\overline{x} \pm t^{*} \frac{s}{\sqrt{n}}$$ $$t=\frac{\overline{x}-\mu_0}{\frac{s}{\sqrt{n}}}$$
Paired Means $$t$$ Sample size is at least 30 OR differences are normally distributed in the population $$\frac{\sigma_d}{\sqrt n}$$ $$\overline{x}_d \pm t^* \left(\frac{s_d}{\sqrt{n}}\right)$$ $$t=\frac{\bar{x}_d-\mu_0}{\frac{s_d}{\sqrt{n}}}$$

* If the population standard deviation is known, then the sampling distribution may be approximated by a $$z$$ distribution.

This lesson may be overwhelming if you are trying to memorize all of the formulas that were introduced. Luckily, doing so is unnecessary. You should focus on seeing the similarities between all of the procedures. For one sample mean, one sample proportion, paired means, two independent means, and two independent proportions, the following general formulas can be applied

General Form of a Confidence Interval
$$sample\ statistic\pm(multiplier)\ (standard error)$$
General Form of a Test Statistic
$$test\;statistic=\frac{sample\;statistic-null\;parameter}{standard\;error}$$

Also, when given raw data, use Minitab Express. It is much more efficient than performing calculations by hand.