Intro. to Risk Analysis, Fall 2001

TO DISCUSS MONDAY AND/OR WEDNESDAY, OCTOBER 8/10

I. Suppose we have used a simple random sample of size n from a certain population to perform a t-test with null hypothesis

and alternate hypothesis

H_a: .

Which of the following statements best expresses the assertion that the p-value is .04:

1. The probability that is .04

2. The probability of obtaining a value of t at least as large as the one we obtained from our sample .04.

3. If it is true that , then the probability of obtaining a value of t (from a sample taken from the population in question) at least as large as the one we obtained from our sample is .04.

4. If it is true that , then the probability of obtaining a value of t (from a simple random sample taken from the population in question) at least as large as the one we obtained is .04.

5. If it is true that , then the probability of obtaining a value of t (from a simple random sample of size n taken from the population in question) at least as large as the one we obtained is .04.

II. Decide which of the following best fits the statement given: Accurate statement, almost accurate, ambiguous (might be correct or incorrect depending on the interpretation), definitely wrong, just gives the intuitive idea, reasonable conclusion, nonsense, etc. In some cases, it may be helpful to think of the distinction between frequentist and subjective probability.

A. Hypothesis tests

1. A hypothesis test allows us to formally test whether a value other than what we have calculate from our sample is likely or unlikely to be the actual value.

2. In a hypothesis test for a mean, the p-value is a measure of the probability that the true mean is accurately approximated by the mean of the sample.

3. The smaller the z-score, the more likely the null hypothesis is correct.

4. Increasing t-values imply decreasing probability that .

5. If the p-value is .03, then there is a 3% chance that the null hypothesis is wrong.

6. The p-value is the probability of observing a value of the test statistic at least as extreme as the one we actually obtained.

7. The p-value is the probability of observing a value of the test statistic at least as extreme as the one we actually obtained, if indeed the null hypothesis is true, assuming that we are only taking simple random samples of the same size from the same population.

8. The p-value is a measure of the weight of the evidence against the null hypothesis, with small values providing evidence against the null hypothesis.

B. Confidence intervals

1. The statistical tool that is used to determine how well we think the sample mean reflects the true mean is the confidence interval, or the probability that an acceptable wide interval around the sample mean encompasses the true mean.

2. If we have calculated a 90% confidence interval (3.49, 4.21) for a mean, then there is a 90% certainty that the range 3.49 to 4.21 bounds the true mean.

3. If we calculate the 95% confidence interval (1.2 , 3.1) for a mean µ, then the probability that 1.2 < µ < 3.1 is 0.95

5. If we have calculated a 90% confidence interval 55 < µ < 58 for a mean µ, then 90% of the time, µ will be between 55 and 58.

6. In calculating a 90% confidence interval 55 < µ < 58 for a mean µ, we use a method which, in repeated sampling with simple random samples, will give an interval containing µ about 90% of the time.

7. In calculating a 90% confidence interval 55 < µ < 58 for a mean µ, we use a method which for about 90% of all possible simple random samples will give an interval containing µ.