Significance tests A test of H0 : p=0.65 against Ha : p<0.65 has test statistic z=-1.78 (a) What

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Significance tests A test of H0 : p=0.65 against Ha : p<0.65...

Significance tests $\mathrm{A}$ test of $H_{0} : p=0.65$ against $H_{a} : p<0.65$ has test statistic $z=-1.78$ (a) What conclusion would you draw at the 5$\%$ significance level? At the 1$\%$ level? (b) If the alternative hypothesis were $H_{a} : p \neq 0.65$ what conclusion would you draw at the 5$\%$ significance level? At the 1$\%$ level?

Significance tests $\mathrm{A}$ test of $H_{0} : p=0.65$ against $H_{a} : p<0.65$ has test statistic $z=-1.78$
(a) What conclusion would you draw at the 5$\%$ significance level? At the 1$\%$ level?
(b) If the alternative hypothesis were $H_{a} : p \neq 0.65$ what conclusion would you draw at the 5$\%$ significance level? At the 1$\%$ level?

Key Concepts

One-Tailed vs Two-Tailed Tests

One-tailed tests are used when the alternative hypothesis specifies a directional difference, meaning the rejection region is in one tail of the distribution. Two-tailed tests are used when the alternative hypothesis only suggests a difference without specifying a direction, distributing the rejection region equally into both tails of the distribution. The choice between these tests changes the critical values and rejection criteria.

Significance Level and Critical Values

The significance level, often denoted by ? (e.g., 5% or 1%), is the threshold used to determine whether the observed data are sufficiently unlikely under the null hypothesis to warrant rejection. This level corresponds to critical values from the test statistic distribution (such as the z-distribution), and if the calculated statistic falls beyond this critical value, the null hypothesis is rejected.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether to reject a statement (the null hypothesis) about a population parameter based on sample data. It involves comparing a test statistic to a critical value, which reflects the probability of observing the data if the null hypothesis were true, thereby enabling an objective decision on the validity of the hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents a default claim or status quo about a population parameter, while the alternative hypothesis (Ha) represents a competing claim. Deciding whether to reject H0 depends on the data collected and the direction specified in Ha (for instance, one-tailed if testing for a specific direction, or two-tailed if testing for any difference).

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