3. A randomly selected group of 12 adults participated in a sleep study. The mean number of hours slept was I = 6.8 hours. The number of hours slept by adults seems to be normally distributed with mean 7 hours and standard deviation 2 hours. a) Describe the sampling distribution of the sample mean I. b) What is the probability of drawing a sample of 12 adults with a mean of 6.8 hours of sleep or less? Give your answer as a percentage rounded to 3 decimal places. 4. In a random sample of 1480 adults, 455 of the respondents have sleepwalked. a) Let p represent the proportion of all adults who have sleepwalked. Find a point estimate for p rounded to 3 decimals places. b) Find 90% and 99% confidence intervals for p, again rounded to 3 decimal places. c) Using your 99% confidence interval from part b), make a conclusion about the results of drawing the sample. d) Describe what happens to the width of the confidence interval as the level of confidence increases. 5. A group of 6 students are asked to estimate the length of one minute without reference to any timers. The times they provided (in seconds) are given below: 53 67 63 55 75 58 Assuming the population is approximately normal and using a 0.10 significance level, test the claim that these provided times are indeed from a population with mean 60 seconds. a) State the null and alternative hypotheses. b) Calculate the test statistic (round it to 3 decimal places). c) Find the critical value (round it to 3 decimal places). d) Make a conclusion based on your answers in parts b) and c). Do the results indicate that students are reasonably good at estimating the length of one minute? 6. A researcher claims that the mean SAT score is greater than 515. A sample of 1800 students shows the sample mean SAT score is 519 with standard deviation 111. Test the researcher's claim using a 0.05 significance level. a) State the null and alternative hypotheses. b) Calculate the test statistic (round it to 3 decimal places). c) Find the P-value (round it to 3 decimal places). d) Make a conclusion based on your answers in parts b) and c). e) Explain what would constitute a Type I error in this context. f) Explain what would constitute a Type II error in this context