College professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X = the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is as follows. f(x) = kx^2 0 ? x ? 20 otherwisea) Find the value of k. (Enter your answer to three decimal places.).375b)What is the probability that the lecture ends within 1 min of the end of the hour? (Enter your answer to three decimal places.) .125c) What is the probability that the lecture continues beyond the hour for between 15 and 45 sec? (Round your answer to four decimal places.)(d) What is the probability that the lecture continues for at least 75 sec beyond the end of the hour? (Round your answer to four decimal places.)

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.