Let P(t)P(t) be the performance level of someone learning a skill as a function of the training time tt. The derivative dPdtdPdt represents the rate at which performance improves. If MM is the maximum level of performance of which the learner is capable, then a model for learning is given by the differential equation

dPdt=k(M−P(t))dPdt=k(M−P(t))

where kk is a positive constant.
Two new workers, Andy and Peter, were hired for an assembly line. Andy could process 1212 units per minute after one hour and 1313 units per minute after two hours. Peter could process 1111 units per minute after one hour and 1414 units per minute after two hours. Using the above model and assuming that P(0)=0P(0)=0, estimate the maximum number of units per minute that each worker is capable of processing.
Andy: ,

According to a research study, parents with young children slept 6.4 hours each night last year, on average. a random sample of 18 parents with young children was surveyed and the mean amount of time per night each parent slept was 6.8. this data has a sample standard deviation of 0.9. (assume that the scores are normally distributed.) researchers conduct a one-mean hypothesis at the 5% significance level, to test if the mean amount of time parents with young children sleep per night is greater than the mean amount of time last year. the null and alternative hypotheses are h0: μ=6.4 and ha: μ> 6.4, which is a right-tailed test. the test statistic is determined to be t0=1.89 using the partial t-table below, determine the critical value(s). if there is only one critical value, leave the second answer box blank.