1 Suppose that a random sample of 50 bottles of a particular

1. Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected. and the alcohol content of each bottle is determined. Let mu denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (8.0, 9.6). a. Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. b. Consider the following statement: There is a 95% chance that p is between 8 and 9.6. Is this statement correct? Why or why not? c. Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 8.0 and 9.6. Is this statement correct? Why or why not? d. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 955 interval ire repeated 100 times. 95 of the resulting intervals will include p. Is this statement correct? Why or why not? 2. A CI is desired for the true average stray-load loss p (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with = 3.0. a. Computea95% Cl for mu when n:25and x=60. b. Compute a 95% Cl for mu when n = 100 and x= 60. c. Compute a 99% Cl for mu when n= 100 and x=60. d. Compute an 82% CI for mu when n= 100 and x=60. e. How large must n be if the width of the 99% interval for mu is to be 1.0? 3. Consider the 1000 95% confidence intervals (Cl) for p that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another. How many of these 1000 intervals do you expect to capture the corresponding value of p? What is the probability that between 950 and 970 of these intervals contain the corresponding value of p? (Hint: Let Y = the number among the 1000 intervals that contain p. What kind of random variable is Y?). 4. A random sample of 100 lightning flashes in a certain region resulted in a sample average radar echo duration of .81 sec and a sample standard deviation of .34 sec. Calculate a 99% (two-sided) confidence interval for the true average echo duration p, and interpret the resulting interval. 5. The superintendent of a large school district, having once had a course in probability and statistics. believes that the number of teachers absent on any given day has a Poisson distribution with parameter A. Use the accompanying data on absences for 50 days to derive a large-sample Cl for A. [Hint: The mean and variance of a Poisson variable both equal so has approximately a standard normal distribution. Now proceed as in the derivation of the interval for p by making a probability statement (with probability 1 - a) and solving the resulting inequalities for A.

Solution

a) Narrower as margin of error is 1.65sigma instead of 1.96 sigma on either side.

b) Not true. THe confidence interval means in the long run we say with 95% confidence that sample mean falls in that interval for large sample sizes.

c) Statement is correct. That is the interpretation of confidence interval

d) This statement is right. As no of trials increses, mu becomes the mean of the conf interval.