Hw 1 Prob 1 A sample of 20 cigarettes is tested to determine nicotine content and the average value observed was 1.2 mg. Compute a 99 percent two-sided confidence interval for the mean nicotine content of a cigarette if it is known that the standard deviation of a cigarette’s nicotine content is σ = 0.2 mg.

Hw 1 Prob 2 In Problem 1, suppose that the population variance is not known in advance of the experiment. If the sample variance is 0.04, compute a 99% two-sided confidence interval for the mean nicotine content. In problem 1, we knew s = 0.2. Now we must estimate s with the sample standard deviation s=.2. Note: this is just slightly larger than C.I. in problem one because we have a fairly large sample. For large n, the difference between the normal and t distributions starts to disappear.

Hw 1 Prob 3 The daily dissolved oxygen concentration for a water stream has been recorded over 30 days. If the sample average of the 30 values is 2.5 mg/liter and the sample standard deviation is 2.12 mg/liter, determine a value which, with 90 percent confidence, exceeds the mean daily concentration. Since n=30, we could choose to use a t-distribution with 29 degrees of freedom or a standard normal distribution. The difference will be small. I will use the standard normal. For a standard normal a confidence interval is given by Note that the way the problem, we are more likely interested only in an upper limit. In this case, we could conduct a one sided confidence interval. In this case, we have

Hw 1 Prob 3 The daily dissolved oxygen concentration for a water stream has been recorded over 30 days. If the sample average of the 30 values is 2.5 mg/liter and the sample standard deviation is 2.12 mg/liter, determine a value which, with 90 percent confidence, exceeds the mean daily concentration. Since n=30, we could choose to use a t-distribution with 29 degrees of freedom or a standard normal distribution. The difference will be small. I will use the standard normal. For a standard normal a confidence interval is given by If we do a one sided C.I. using the t distribution with 29 degrees of freedom, we get

Hw 1 Prob 4 Suppose that when sampling from a normal population having an unknown mean µ and unknown variance σ2, we wish to determine a sample size n so as to guarantee that the resulting 100(1 – α) percent confidence interval for µ will be of size no greater than A, for given values α and A. Explain how we can approximately do this by a double sampling scheme that first takes a subsample of size 30 and then chooses the total sample size by using the results of the first subsample. Since we do not know s until we collect data, we cannot know the t value to look up. Iterative solution 1. Estimate a sample size n 2. Collect data and calculate s 3. If we have achieved the desired precision, stop. If not, calculate a sample size based on formula above. 4. Collect additional data and calculate new s. 5. Go to step 3.