Final answer:
In statistical hypothesis testing, you formulate a null hypothesis (H o), which often posits no effect or difference, and an alternative hypothesis (Ha). A test statistic is compared to a critical value from the z or t-distribution to determine if the null hypothesis should be rejected, based on the level of significance (alpha) and p-value.
Step-by-step explanation:
Hypothesis Testing in Statistics
In statistical hypothesis testing, you compare an observed sample statistic to a parameter value stated in the null hypothesis. The null hypothesis, denoted as H o, typically represents no effect or no difference, and the alternative hypothesis, denoted as Ha, represents the effect or difference you are testing for. The random variable in the context of this question could be the difference in sample proportions or means, depending on whether it's a test of two proportions or two means.
If you are conducting a test at the 5% significance level, your alpha (α) is 0.05. This is the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. The p-value indicates the probability of observing a statistic as extreme as the test statistic if the null hypothesis is true. If the p-value is less than α, you reject the null hypothesis; if it is greater, you fail to reject it.
For comparing two independent population proportions, such as the proportion of homes with cable television service in different communities, you would use a z-distribution provided the sample size is sufficiently large to approximate normality under the Central Limit Theorem. The pooled proportion is needed to compute the standard error of the difference between two sample proportions.
Concerning the unknown population standard deviations and the comparison of means, the t-distribution is used to perform the test. The conclusions from the hypothesis test depend on the comparison of the p-value to the alpha level and whether you are using a one-tailed or two-tailed test.