One-Sample Hypothesis Test for Proportions: Learn It 1

  • Complete a one-sample [latex]z[/latex]-test for proportions from hypotheses to conclusions.
  • Use a P-value to explain the conclusions of a completed [latex]z[/latex]-test for proportions.

One-Sample Hypothesis Test ([latex]z[/latex]-test) for Proportions

Hypothesis testing is part of inference.

In inference, we use a sample to draw a conclusion about a population. So, the purpose of a hypothesis test is to use sample data to test a claim about a population parameter.

From the claim, we state an assumption about the value of the population parameter. Could the data have come from this population? Or is the sample proportion too far off? It depends on how much random samples from this population vary. We make a judgment about whether the sample proportion is likely or unlikely to occur based on a sampling distribution for the parameter. If the data supports our claim and is unlikely, then we doubt our assumption about the population proportion.

Let’s conduct a hypothesis test!

In a hypothesis test, the first step is to clearly state the null hypothesis ([latex]H_{0}[/latex]) and the alternative hypothesis ([latex]H_{A}[/latex]).

The null hypothesis, [latex]H_{0}[/latex], is what we assume to be true to begin with. This can often be considered the status quo, and as a result, if you cannot accept the null, it requires some action.

  • For proportion, our parameter is population proportion, denoted [latex]p[/latex], therefore our null hypothesis is [latex]H_0: p = \text{null value}[/latex]

The alternative hypothesis, [latex]H_{A}[/latex], is what we consider to be plausible if the null hypothesis is false. This is usually what the researcher is trying to prove.

  • The alternative hypothesis can be:
    • [latex]H_{A}: p>\text{null value}[/latex],
    • [latex]H_{A}: p<\text{null value}[/latex], or
    • [latex]H_{A}: p \neq \text{null value}[/latex].
The second step in a hypothesis test is to check whether the conditions or assumptions for the test have been met. 
Conditions for the hypothesis test.

For testing a one-sample [latex]z[/latex]-test for proportions, we require:

  • Large counts: Check that [latex]np\ge10[/latex] and [latex]n(1-p)\ge10[/latex].
  • Random samples/assignment: Check that the sample is a random sample.
  • 10% population size: Check that the sample size, [latex]n[/latex], is less than 10% of the population size, [latex]N[/latex]: [latex]n<0.10(N)[/latex]

Internet Access

A young child using a computer to browse the Internet

The coronavirus outbreak has driven many commercial and social activities online, and for some, the internet has become an ever more crucial link to those they love and the things they need. As Americans turn to the internet for critical purposes, there are rekindled debates about how the digital divide – that is, the gap between those who do or do not have access to technology – may hinder people’s ability to complete everyday tasks or even schoolwork.[1]

  1. https://www.pewresearch.org/internet/2020/04/30/53-of-americans-say-the-internet-has-been-essential-during-the-covid-19-outbreak/ ↵