D.25 Answers: Tests for one mean
Answers to exercises in Sect. 27.13.
Answer to Exercise 27.1:
1. H0: μ=7725; H1: μ≠7725 (two tailed).
2. ˉx=6753.64 and s.e.(ˉx)=s/√n=1142.123/√11=344.363.
3. t=(6753.64−7725)/344.363=−2.821, as in output.
This ‘large’; expect small P-value; software confirms this: two-tailed P=0.018.
4. Moderate evidence (P=0.018) that the mean energy intake
is not meeting the recommended daily energy intake
(mean: 6753.6kJ; std. dev.: 1142.1kJ).
Answer to Exercise 27.2:
H0: μ=120 and H1: μ≠120 (two-tailed), where μ is the mean time in seconds.
Standard error: s.e.(ˉx)=23.8/√85=2.581472.
t-score: (60.3−120)/2.581472=−23.13,
which is huge; P-value will be really small.
Very strong evidence (P<0.001) that children do not spend 2 minutes (on average)
brushing their teeth (mean: 60.3s; std. dev.: 23.8s).
Answer to Exercise 27.3:
H0: μ=50 and H1: μ>50 (one-tailed), where μ is the mean mental demand.
Standard error: s.e.(ˉx)=22.05/√22=4.701076.
t-score: (84−50)/4.701076=7.23,
which is very large; P-value will be very small.
Very strong evidence (P<0.001) that the mean mental demand is greater than 50.
(Notice we say greater than, because of the RQ and the alternative hypothesis.)
Answer to Exercise 27.4:
Physical:* t=−1.28;
Mental:* t=1.80.
The P-values both larger than 5%.
No evidence that the mean score for patients is different than the general population score.
Answer to Exercise 27.5:
H0: μ=12 and H1: μ≠12 (two-tailed), where μ is the mean weight in grams.
Standard error: s.e.(ˉx)=0.60652/√43=0.09249343.
t-score: (14.9577−12)/0.09249343=31.98,
which is huge; P-value will be very small.
Very strong evidence (P<0.001) that the mean weight of a Fun Size Cherry Ripe bar
is not 12 grams (mean: 14.9577; std. dev.: 0.067g),
and they may be larger.
Answer to Exercise 27.6:
H0: μ=1000 and H1: μ≠1000,
where μ is the population mean guess of the spill volume.
Standard error: 46.15526.
t-score: (846.4−1000)/46.15526=−3.33,
which is very large (and negative),
so the P-value will be very small.
Very strong evidence that the mean guess
of blood volume is not 1000,ml, the actual value.
The sample is much larger than 25:
the test is statistically valid.
Answer to Exercise 27.7: Hypotheses have the form H0: μ=pre-determined target, and H1: μ≠pre-determined target. t-scores: t1=0.318, t2=2.347, t3=−0.466, t4=−0.726. P-values will be large, except for second test. No evidence that the instruments are dodgy, except perhaps for the first instrument for mid-level LH concentrations. Should be statistically valid.
While assessing the means is useful, how variable the measurements are is also useful (but beyond us).