Problem 6: Spider silk is the strongest known material, natural or human-made, on a weight basis. A study examined the mechanical properties of spider silk using a random sample of 21 female golden orb weavers, Nephila calvipes. The natural logarithms of silk-stress yield (in megapascals, MPa) which represents the amount of force per unit area needed to achieve permanent deformation of the silk strand are as follows: 5.10, 6.17, 5.53, 5.86, 5.15, 6.11, 5.71, 5.89, 5.61, 6.61, 5.80, 5.79, 5.60, 5.81, 5.67, 5.17, 5.64, 5.46, 5.88, 5.60, 5.67 (Brigitte Baldi and David S. Moore, 2018, The Practice of Statistics in the Life Sciences, p. 441). The sample mean and standard deviation are 5.7062 and 0.3467, respectively.
Q1. Construct a 95% confidence interval for the population mean log silk-stress yield and write a brief interpretation of your confidence interval in terms of the context.
Q2. Under what assumption on the distribution of the population log silk-stress yield is the confidence interval in Q1 constructed?
Q3. Without constructing a 99% confidence interval, what can you say about the 99% confidence interval compared to the 95% one in Q1 in terms of the interval length? Write a sentence to justify your answer.
Q4. Do the data support the claim that the population mean log silk-stress yield is larger than 5.5 at a significance level of 0.05? Answer the question by the following steps.
a) Formulate null and alternative hypotheses in terms of the population parameter as well as the context of this application;
b) State type I and type II errors in terms of the context of this application.
c) Is the test right-tailed, left-tailed or two-tailed?
d) Find the P-value of the test, interpret your P-value in the context of the application, and make your conclusion.