## 28.12 Exercises

Selected answers are available in Sect. D.26.

Exercise 28.1 Use the 68–95–99.7 rule to approximate the two-tailed $$P$$-value if:

1. the $$t$$-score is $$3.4$$.
2. the $$t$$-score is $$-2.9$$.
3. the $$t$$-score is $$1.2$$.
4. the $$t$$-score is $$-0.95$$.
5. the $$t$$-score is $$-0.2$$.
6. the $$t$$-score is $$6.7$$.
Exercise 28.2 Consider the $$t$$-scores in Exercise 28.1. Use the 68–95–99.7 rule to approximate the one-tailed $$P$$-values in each case.
Exercise 28.3 Suppose a hypothesis test results in a $$P$$-value of 0.0501. What would we conclude? What about if the $$P$$-value was 0.0499?

Exercise 28.4 Consider again the study to determine the mean body temperature, where $$\bar{x} = 36.8051^{\circ}\text{C}$$. What, if anything, is wrong with these hypotheses? Explain.

1. $$H_0$$: $$\bar{x} = 36$$ and $$H_1$$: $$\bar{x} \ne 36$$.
2. $$H_0$$: $$\bar{x} = 36.8051$$ and $$H_1$$: $$\bar{x} > 36.8051$$.
3. $$H_0$$: $$\mu = 36.8051$$ and $$H_1$$: $$\mu \ne 36.8051$$.
4. $$H_0$$: $$\mu = 36$$ and $$H_1$$: $$\mu = 36.8051$$.
5. $$H_0$$: $$\mu > 36$$ and $$H_1$$: $$\bar{x} > 36$$.
6. $$H_0$$: $$\mu = 36$$ and $$H_1$$: $$\mu > 36$$.

Exercise 28.5 The recommended daily energy intake for women is 7725kJ (for a particular cohort, in a particular country; ). The daily energy intake for 11 women was measured to see if this is being adhered to. The RQ was

Is the population mean daily energy intake 7725kJ?

The test produced $$P=0.018$$. What, if anything, is wrong with these conclusions after completing the hypothesis test?

1. There is moderate evidence ($$P = 0.018$$) that the energy intake is not meeting the recommended daily energy intake.
2. There is moderate evidence ($$P = 0.018$$) that the sample mean energy intake is not meeting the recommended daily energy intake.
3. There is moderate evidence ($$P = 0.018$$) that the population energy intake is not meeting the recommended daily energy intake.

Exercise 28.6 A study compared ALDI batteries to another brand of battery. In one test comparing the length of time it takes for 1.5 volt AA batteries to reach 1.1 volts, the ALDI brand battery took 5.73 hours, and the other brand (Energizer) took 5.44 hours .

1. The $$P$$-value for comparing these two means is about $$P=0.70$$. What does this mean?
2. Is this difference likely to be of any practical importance? Explain.
3. What would be a useful, but correct, conclusion for ALDI to report from the study? Explain.
4. What else would be useful to know in comparing the two brands of batteries?

### References

Altman DG. Practical statistics for medical research. Chapman & Hall; 1991.
Dunn PK. Comparing the lifetimes of two brands of batteries. Journal of Statistical Education. 2013;21(1).