MATH 2565 - Winter 2017 Assignment 3 due March 14 1. In a survey conducted by the Big Creek State Marine Reserve in California, the length of various species of fish caught at two locations were recorded. Suppose the length of a yellowtail caught at the Santa Lucia Kelp Bed is normally distributed with mean 38 inches and standared deviation 2.9 inches. a. Suppose 1 yellowtail is selected at random. What is the probability that the length of the yellowtail is more than 41 inches? b. Suppose 32 yellowtail are selected at random. What is the probability that the sample mean length is greater than 41 inches? 2. Suppose a random sample of size n=1000 is obtained from a population with probability of success p=0.25. a. Find the value a such that P (P a) = 0.05. b. Find the value c such that P (0.25 c P 0.25 + c) = 0.99. c. Find the quartiles of the distribution of P . 3. A random sample of professional wrestlers was obtained, and the annual salary(in dollars) for each was recorded. The summary statistics were x = 47, 500 and n = 18. Assume the distribution of annual salary is normal with = 8, 500. a. Find a 90% confidence interval for the true mean annual salary for all professional wrestlers. b. How large a sample is necessary in order for the bound on the error of estimation to be 3000? 4. A typical washing machine has several different cycles, including soak, wash, and rinse. The energy consumption of a washing machine is linked to the length of each cycle. A random sample of 21 washing machines was obtained, and the length(in minutes) of each wash cycle was recorded. The summary statistics are x = 37.8 and s = 5.9. Assume the underlying distribution of main wash cycle times is normal. a. Find a 90% confidence interval for the true mean wash cycle time. b. Interpret the confidence interval found in part (a). c. Suppose a 95% confidence interval for the true mean wash cycle time is constructed. Would this interval be smaller or larger than the interval in part (a)? Justify your answer. 5. There are three factors that affect the width of a large-sample confidence interval for p: the confidence interval, the sample size, and the sample proportion, p. In each of the following problems, determine whether the width of the resulting confidence interval for p increases or decreases. a. Confidence level and p are constant, and n increases. b. Sample size and p are constant, and confidence level decreases. c. Sample size and confidence level are constant, and p(>0.5) increases. 1