The goal of inferential statistics is to infer information about an entire population by examining a random sample from that population.

Testing a Claim about a Population Proportion Intuitively

• In statistics a hypothesis is a statement or claim about a property of an entire population.

  • In this section we study hypotheses about the true proportion p of individuals in the population with a particular characteristic.

  • Suppose that individuals having a particular property are assumed to form a proportion p0 of a population. To answer the question of how this assumption compares to the true proportion p of these individuals, we begin by stating two opposing hypotheses.

• The null hypothesis, denoted by H0, states the “assumed state of affairs” and is expressed as an equality: p=p0

• The alternative hypothesis (also called the research hypothesis), denoted by H1, is the proposed substitute to the null hypothesis and is expressed as an inequality: p>p0, or p<0, or p≠p0.

  • The P-value is the probability that the difference between the proportions in the two samples is due to chance alone. So if the P-value is small (less than the significance level of the test), we reject the null hypothesis.

Testing a Claim about a Population Proportion Using Probability: The -Value

• A “very small” P-value tells us that it is “very unlikely” that the sample we got was obtained by chance alone, so we should reject the null hypothesis.

• The P-value at which we decide to reject the null hypothesis is called the significance level of the test and is denoted by alpha (α).

Testing a Hypothesis

1. Formulate the Hypotheses. First state a null hypothesis and an alternative hypothesis.

2. Obtain a Random Sample. Obtain a sample that is selected arbitrarily and without bias.

3. Calculate the P-Value. Use a calculator to find the p-value associated with the sample obtained.

4. Make a Conclusion. A p-value smaller than the significance level of the test indicates that we should reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Inference about Two Properties

• Most statistical studies involve a comparison of two groups, usually called the treatment group and the control group.

  • For example, when researchers are testing the effectiveness of an investigational medication, two groups of patients are selected.

  • The individuals in the treatment group are given the medication; the individuals in the control group are not. The proportions of patients who recover in each group is compared.

  • The goal of the study is to determine whether the difference in the proportion that recovers is statistically significant (and not just the result of chance alone).