Preparing for the next class
In the next in-class activity, you will need to be able to conduct a complete hypothesis test for a proportion and write the conclusion of a hypothesis test in context. You will also need to understand that there are limitations on P-values and verify that the conditions of the one-sample z-test for proportions have been met.
In the previous in-class activity, you learned that a P-value is a probability that measures the likelihood of observing a test statistic at least as extreme as the one observed (in the direction of the alternative hypothesis). Also, recall that once a P-value is calculated, we compare it to the significance level in order to decide whether we have enough evidence to reject the null hypothesis in favor of the alternative hypothesis.
Ultimately, when we calculate and use the P-value, the goal is to arrive at the conclusion of a hypothesis test. Once we decide whether we have enough evidence to reject a null hypothesis, we write a statement in context of the original question asked in order to describe the outcome of the hypothesis test. It is important to remember that we never prove that a null hypothesis is true; we only conclude that the sample data collected either do or do not support the alternative hypothesis. Therefore, one of two statements will be concluded, as presented in the table below.
| Decision | Conclusion |
| If P-value [latex]\leq \alpha[/latex], there is enough evidence to reject the null hypothesis. | At the [latex]\alpha[/latex] × 100% significance level, the data provide convincing evidence in support of the alternative hypothesis. |
| If P-value [latex]> \alpha[/latex] there is not enough evidence to reject the null hypothesis. | At the [latex]\alpha[/latex] × 100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis. |
Question 1
According to a 2003 Pew Research study, 70% of college students reported playing video games.[1] A researcher wonders if more than 70% of college students play video games today. She collects data from a random sample of college students. Consider the hypotheses:
[latex]H_{0}:p = 0.7H_{A}:p > 0.7[/latex]
Based on her sample, she obtains a test statistic and a P-value of 0.024.
- Is there reason to reject the null hypothesis at a 5% significance level? Explain.
Hint: Recall the rejection rule: If the P-value is less than or equal to [latex]\alpha[/latex], reject the null hypothesis. - Fill in the blank with either “do” or “do not.”
At the 5% significance level, the data _____ provide convincing evidence that more than 70% of college students play video games. - Is there reason to reject the null hypothesis at a 1% significance level? Explain.
Hint: Recall the rejection rule: If the P-value is less than or equal to [latex]\alpha[/latex], reject the null hypothesis. - Fill in the blank with either “do” or “do not.”
At the 1% significance level, the data _____ provide convincing evidence that more than 70% of college students play video games.
To summarize, in order to conduct a hypothesis test, you need to do the following:
One-Sample Z-Test of Proportions *MISSING LATEX*
- Write out the null and alternative hypotheses.
- Check the conditions for the hypothesis test. For testing a one-sample z-test for proportions, we require:
- Large Counts: Check that 10 and 10.
- Random Samples/Assignment: Check that the sample is a random sample.
- 10% Population Size: Check that the sample size, , is less than 10% of the population size, : 0.10.
- Calculate a test statistic.
- Calculate a P-value.
- Compare the P-value to the significance level to make a decision.
- Write a conclusion in context (e.g., we do/do not have convincing evidence…).
Question 2
A student reads a headline claiming “20% of college students in the United States have student loans.” She believes the percentage of students in the United States who have student loans is much higher. She polls 15 of her friends and asks them “Do you have student loans?” She wants to prove at the 1% significance level that the percentage of students in the United States who have student loans is higher than 20%. Are the conditions of the one-sample z-test for proportions met?
Hint: Check the three conditions for the one-sample proportion z-test.
Question 3
A Statista Global Consumer Survey reports that approximately 20% of people in Nigeria say they own some type of cryptocurrency, such as Bitcoin, Ethereum, or Dogecoin.[2] A researcher in the United States believes to have observed hesitancy of Americans to purchase cryptocurrency. He obtains a random sample of 850 Americans and asks if they have purchased cryptocurrency. Of those surveyed, 142 replied that they have. At the 5% significance level, do the data provide convincing evidence that the proportion of Americans buying cryptocurrency is less than what is reported in Nigeria?
- Write the null hypothesis using the appropriate symbolic notation.
- Write the alternative hypothesis using the appropriate symbolic notation.
- Verify the conditions of the one-sample z-test for proportions.
Hint: There are three conditions to check. Remember that [latex]p[/latex] is the null hypothesized value for the large count condition. - Calculate the test statistic using the DCMP Inference for Population Proportion tool that was introduced for confidence intervals in In-Class Activity 10.D.
- Go to https://dcmathpathways.shinyapps.io/Inference_prop/.
- Under “Enter Data,” select “Number of Successes.”
- Enter “Sample Size” and “# of Successes.”
- Add appropriate labels for success and failure by selecting “Success/Failure.”
- Under “Type of Inference,” select “Significance Test.”
- Enter the “Null Value of [latex]p_{0}[/latex].” This is the hypothesized value in the null hypothesis.
- Select the appropriate “Alternative” given your alternative hypothesis.
- Remember that:
- “Two-sided” corresponds to [latex]H_{A}: p \neq p_{0}[/latex].
- “Less” corresponds to [latex]H_{A}: p < p_{0}[/latex].
- “Greater” corresponds to [latex]H_{A}: p > p_{0}[/latex].
- What is the P-value?
Hint: Remember that you have a lower-tailed test. - Will the null hypothesis be rejected? Explain.
- At the 5% significance level, do the data provide convincing evidence that the proportion of Americans buying cryptocurrency is less than what is reported in Nigeria? Write the conclusion in a sentence.
A note on P-values: Though P-values are widely used, there are some limitations on their use and significant debate as to their reliability. Recall that a P-value is calculated based on one sample of data collected, and, as a result, it may be difficult to obtain a similar P-value upon replication of an experiment. Additionally, P-values can be manipulated by increased sample size. In some studies, instead of using a P-value to answer the binary question (“Is there or is there not statistical significance?”), it may be better to consider the effect size which answers the question “How strong is the effect in the sample?” For example, in reporting the effect size, a researcher explains by how much a treatment works rather than just if it works. Additionally, in using effect sizes, quantitative comparisons between the results of different studies can be made.
Question 4
Determine whether this statement is true or false: A P-value explains how strong an effect is in a sample of data.
Looking ahead
In the next class, you will be revisiting the results of a research study done at Virginia Tech to analyze the condition of plumbing in Flint, Michigan. Over the last several years, Flint, Michigan has been in the news because of contaminated water. To get a better idea of what has occurred in Flint, Michigan, visit https://www.nrdc.org/stories/flint-water-crisis-everything-you-need-know. Read about the problems that have occurred because of lead-contaminated water here: https://www.nrdc.org/stories/flint-water-crisis-everything-you-need-know#sec-whyis.
You should be able to:
- Briefly describe the situation.
- Identify effects of drinking lead-contaminated water.
