D.25 Answers: Tests for one mean

Answers to exercises in Sect. 27.13.

Answer to Exercise 27.1: 1. H0: μ=7725; H1: μ7725 (two tailed). 2. ˉx=6753.64 and s.e.(ˉx)=s/n=1142.123/11=344.363. 3. t=(6753.647725)/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: μ120 (two-tailed), where μ is the mean time in seconds. Standard error: s.e.(ˉx)=23.8/85=2.581472. t-score: (60.3120)/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: (8450)/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: μ12 (two-tailed), where μ is the mean weight in grams. Standard error: s.e.(ˉx)=0.60652/43=0.09249343. t-score: (14.957712)/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: μ1000, where μ is the population mean guess of the spill volume. Standard error: 46.15526. t-score: (846.41000)/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: μpre-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).