D.26 Answers: More about hypothesis tests
Answers to exercises in Sect. 28.12.
Answer to Exercise 28.1:
Using the 68–95–99.7 rule:
1. Very small; certainly less than 0.003 (99.7% between -3 and 3).
2. Very small; bit bigger than 0.003 (99.7% between -3 and 3).
3. Large-ish: Between 0.32 (68% between 1 and -1) and
5% (95% between -1 and 1), but closer to 0.32.
4. Bit smaller than 0.32 (68% between -1 and 1).
5. Very large! Almost 0.50.
6. Very small; much smaller than 0.003.
Answer to Exercise 28.2:
The answers are half the values given to
Exercise 28.1.
Using the 68–95–99.7 rule:
1. Very small; certainly less than 0.0015 (99.7% between -3 and 3).
2. Very small; bit bigger than 0.0015 (99.7% between -3 and 3).
3. Large-ish: Between 0.16 (68% between 1 and -1) and
2.5% (95% between -1 and 1), but closer to 0.16.
4. Bit smaller than 0.16 (68% between -1 and 1).
5. Very large! Almost 0.25.
6. Very small; much smaller than 0.0015.
Answer to Exercise 28.3: Using the 68–95–99.7 rule, the \(P\)-value is just under 0.05, and hence ‘small’ If the \(t\)-score was 0.0499, the \(P\)-value would be just larger than 0.05 and hence ‘big.’
The difference between 0.0501 and 0.0499 is trivial though… it is silly to jump from ‘evidence supports the alternative hypothesis!’ to the complete opposite conclusion ‘evidence doesn’t support the alternative hypothesis!’ over such a minor difference.
Answer to Exercise 28.4:
1. Hypotheses are about population parameters like \(\mu\),
not sample statistics like \(\bar{x}\).
2. Hypotheses are about parameters like \(\mu\),
not statistics like \(\bar{x}\).
The value of 36.8051 is a sample mean,
but hypothesis are meant to be written before the data are collected.
In any case,
these hypotheses are asking to test if the sample mean is 36.8051…
which we know it is.
3. 36.8051 is a sample mean,
but hypothesis are meant to be written down before the data are collected.
4. 36.8051 looks like a sample mean,
but hypothesis are meant to be written down before the data are collected.
5. Hypotheses are about parameters like \(\mu\), not statistics like \(\bar{x}\).
6. This would be fine, if the RQ was one-tailed… but it is two-tailed.
Answer to Exercise 28.5:
1. The conclusion is about the mean energy intake
(population mean energy intake specifically).
2. Conclusions are never about sample statistics. We want to know what the statistic that tells us about the population parameter 3. The conclusion is about the population mean energy intake.
2. Conclusions are never about sample statistics. We want to know what the statistic that tells us about the population parameter 3. The conclusion is about the population mean energy intake.
Answer to Exercise 28.6:
1. Slight evidence of a difference in lifetime between the two brands.
2. No. The difference is 0.29 hours, or about 17 minutes.
A difference of 17 minutes in over 5 hours of use is trivial.
3. Conclusion: little evidence of a difference between the mean lifetimes.
That’s cumbersome for advertising.
A common advertising trick:
“No other battery lasts longer!”…
meaning there is no evidence of a difference in means.
4. Price!