Raising the bar

Chapter 9. Hypothesis Testing

Unit 9.7. Questions for repetition

$latex \quad$1. In the context of a university admission test discuss the trade-off between Type I and Type II errors.

$latex \quad$2. Explain the statistical logic behind the decision rule using the example of a cereal package (the choice of hypotheses and the use of the confidence interval).

$latex \quad$3. $latex z$ statistics versus $latex p$-values.

$latex \quad$$latex \quad$(a) Give economic examples of the null and alternative hypotheses which lead to (i) a one-tail test and (ii) a two-tail test.

$latex \quad$$latex \quad$(b) How are the critical values defined and what are the decision rules in these cases using $latex z$ statistics?

$latex \quad$$latex \quad$(c) Define $latex p$-values for each of those cases and explain the benefits or drawbacks of applying $latex z$ statistics versus $latex p$-values.

$latex \quad$4. Normal population versus Bernoulli.

$latex \quad$$latex \quad$(a) Select two pairs of hypotheses such that in one case you would apply a two-tail test and in the other $latex -$ a one-tail test. Assuming that the population is normal and the variance is unknown, illustrate graphically the procedure of testing the mean.

$latex \quad$$latex \quad$(b) What changes would you make if the population were Bernoulli?

$latex \quad$5. Using the example of testing the sample mean describe in full the procedure of assessing the power of a test. Answer the following questions:

$latex \quad$$latex \quad$(a) What happens to the power when $latex \mu^*$ moves away from $latex \mu_0$?

$latex \quad$$latex \quad$(b) What happens to $latex 1 - \beta$ when $latex \alpha$ decreases?

$latex \quad$$latex \quad$(c) How does the sample size affect the power?

$latex \quad$$latex \quad$(d) How does $latex 1 - \beta$ behave if $latex \sigma$ increases?

$latex \quad$6. Give the decision tree for testing the difference between two population means distinguishing between:

$latex \quad$$latex \quad$(a) dependent and independent samples,

$latex \quad$$latex \quad$(b) cases of known and unknown variances,

$latex \quad$$latex \quad$(c) equal and unequal variances when they are unknown and

$latex \quad$$latex \quad$(d) normal and Bernoulli populations.

$latex \quad$7. In one block define and describe properties of chi-square, $latex t$ and $latex F$ distributions.

$latex \quad$8. Testing for equality of variances.

$latex \quad$$latex \quad$(a) What is the distribution of the ratio of two sample variances from two normal populations?

$latex \quad$$latex \quad$(b) How would you test for equality of variances from two normal populations?

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