e-Statistics

Guide to Worksheet 5-6

Worksheet No.5. Chapter 5 is vastly different from what we have conceived of ``data exploration'' in Chapter 3. Here we are engaged in ``statistical analysis,'' and introduced to the concept of hypothesis test--null and alternative hypotheses, type I and II errors, test statistics and critical region.

The confidence interval is reasonably reliable when the size of data is larger than 30 without any further assumption [2a].
When the size of data is smaller than 30, it is necessary to check whether the sample distribution follows a normal distribution quite well [2b].
Type II error is the failure to detect that the claim (i.e., alternative hypothesis) is true (when it is true). In the context of Problem 3, it is the failure to see that the new treatment has improved the exercise capacity (when it actually does so).
Critical region is the set of criteria where the test statistic "T" should satisfy when null hypothesis is rejected. In Problem 3 it is "T > 1.662".
The recommended sample size must be integer greater than the calculated value. In Problem 4, it is 135. In Problem 5, it is 46.

Worksheet No.6. The construction of hypothesis test and the interpretation of p-value are the two of the most important skills covered in this worksheet.

Claim	Hypotheses	p-value	Conclusion
The mean comprehension is greater than 80	$H_0: \mu \le 80$ vs. $H_A: \mu > 80$	0.205	No evidence to support the claim
The manufacturer's claim is false	$H_0: \mu \ge 35,000$ vs. $H_A: \mu < 35,000$	0.008	Strong evidence to support the consumer agency's claim
There is a gain in mileage from the device	$H_0: \mu \le 0$ vs. $H_A: \mu > 0$	0.2	Not enough evidence to support the claim (see 1 below)
The mean dissolution rate is less than 20mg	$H_0: \mu \ge 20$ vs. $H_A: \mu < 20$	0.03	Moderate evidence to support the claim (see 2 below)
The average SO2 emissions are less than 0.145	$H_0: \mu \ge 0.145$ vs. $H_A: \mu < 0.145$	0.004	Strong evidence to support the claim

We also learned that a confidence interval accompanied by the hypothesis test must be consistent with the test result [2g]

In Problem 3 we found the probability of type II error to be 0.8, indicating the possibility that we failed to reject incorrectly. Thus, we suggest further study with larger sample size [3fg]. In the future experiment the choice of a larger sample size is the important factor to decide in order to achieve a smaller probability of type II error.
In Problem 4 with the choice of significance level 0.01 we cannot support the claim. The probability of type II error is 0.03, suggesting that we should have been able to reject if the assumption for the true dissolution rate (19.6mg) is correct. Thus, we do not recommend further study [4f].