Question

According to Americans, access to healthcare and the cost of healthcare remain the most urgent health problems. However, a recent Gallup poll \(^{11}\) shows that concern about substance abuse jumped from \(3 \%\) to \(14 \%\) in 2017 . Based on samples of size 200 for each year, the data that follow reflect the results of that poll. $$ \begin{array}{lrr} \hline \text { Concern } & 2016 & 2017 \\ \hline \text { Access } & 40 & 48 \\ \text { Cost } & 54 & 32 \\ \text { Substance abuse } & 6 & 28 \\ \text { Cancer } & 24 & 22 \\ \text { Obesity } & 16 & 14 \\ \text { Other } & 60 & 56 \\ \hline \text { Total } & 200 & 200 \\ \hline \end{array} $$ a. Calculate the proportions in each of the categories for 2016 and 2017 . Test for a significant change in proportions for the healthcare concerns listed from 2016 to 2017 using \(\alpha=.05\) b. How would you summarize the results of the analysis in part a? Can you conclude that the change in the proportion of adults whose concern was substance abuse is significant? Why or why not?

Short answer

Answer: [Insert the decision from part A: Reject or fail to reject the null hypothesis]. Based on the results of the Chi-Squared test, with a test statistic of [Insert Chi-Square test statistic] and a critical value of [Insert critical value], we [reject/fail to reject] the null hypothesis of no significant change in proportions for healthcare concerns listed from 2016 to 2017. Therefore, the increase in the proportion of adults whose concern was substance abuse from 2016 to 2017 is [significant/not significant]. The reasoning behind our conclusion is that if there was a significant change in proportions overall for healthcare concerns, this would also indicate a significant change in the proportion of adults concerned about substance abuse.

Step-by-step solution

Part A: Calculating Proportions and Test for Significant Change

To calculate the proportions for each healthcare concern in each year, we need to divide the number of responses in each category by the total number of responses for that year. Step 1: Calculate proportions for each healthcare concern in 2016 and 2017 $$ \text{Proportion} = \frac{\text{Number of Responses in each Category}}{\text{Total Number of Responses for each year}} $$ Step 2: Perform the test for significant change in proportions using a Chi-Squared test - Establish the null hypothesis, \(H_0\): There is no significant change in proportions for the healthcare concerns listed from 2016 to 2017. - Establish the alternative hypothesis, \(H_A\): There is a significant change in proportions for the healthcare concerns listed from 2016 to 2017. - Calculate the expected counts for each category under the assumption of the null hypothesis. - Compute the Chi-Squared test statistic: $$ \chi^2 = \sum \frac{(\text{Observed Count} - \text{Expected Count})^2}{\text{Expected Count}} $$ - Determine the critical value for the Chi-Squared test using \(\alpha=0.05\) and the appropriate degrees of freedom. - Compare the calculated test statistic to the critical value and make a decision regarding the null hypothesis.

Part B: Summarizing Results and Concluding

To summarize the results of the analysis in part A, we will first report the calculated proportions for each healthcare concern in both years. Then, we'll discuss the results of the Chi-Squared test, including the test statistic, critical value, and whether we reject or fail to reject the null hypothesis. Finally, we'll conclude whether the change in the proportion of adults whose concern was substance abuse is significant based on the results of the analysis and explain the reasoning behind our conclusion.

Key concepts

Calculating Proportions

When we talk about calculating proportions, we're referring to the process of determining what fraction or percentage a particular subgroup represents out of a whole. It is a fundamental step in many statistical analyses, especially when trying to understand the composition of a dataset or the changes over time.

In the case of the healthcare concerns data from the Gallup poll, we calculate the proportion of each concern by dividing the number of people who mentioned a specific concern by the total number of survey respondents for that year. For example, to find the 2016 proportion for Access concerns, we'd divide 40 by 200, yielding 0.20 or 20%. This simple calculation allows us to see the relative importance or urgency of different health concerns amongst the survey participants and to compare these priority levels across different years.

Test for Significant Change

When performing a test for a significant change, we are essentially evaluating whether the observed differences in our data occurred by chance or reflect a true change in the population. In the context of our Gallup poll example, we want to know if the increase in concern for substance abuse, from 3% to 14%, represents a real shift in public concern.

The Chi-Squared test is a commonly used statistical method for this purpose. It compares the frequency of observed results with the frequencies we would expect to find if there were no actual change (the null hypothesis). By analyzing the difference between the observed and expected frequencies, the test can advise us on whether the observed changes are statistically significant. A significant result means that we have enough evidence to reject the null hypothesis and accept that a real change has occurred.

Healthcare Concerns Statistics

Statistics on healthcare concerns often focus on identifying the issues that are most important to a population. This information can guide policymakers and healthcare providers in allocating resources and determining which areas require the most attention.

In the context of the Gallup poll results, we see specific numerical representations of these community concerns. By using statistics to understand and interpret these numbers, stakeholders can make informed decisions regarding healthcare strategy. This may include ramping up measures to combat substance abuse if the increase to 14% in public concern reflects a wider crisis or societal shift in attitudes towards this issue.

Null Hypothesis in Statistics

The null hypothesis in statistics is a default claim that there is no effect or no difference. It's what you test against when you're assessing the likelihood of the alternative hypothesis being true.

When applying this to our Gallup poll exercise, the null hypothesis is that the proportion of concern for each healthcare issue has not significantly changed from 2016 to 2017. To refute this assertion, we use the Chi-Squared test which helps us determine whether the changes in proportions we observe are due to random variation or if they are statistically significant. If we find that the changes are indeed significant, we can reject the null hypothesis and accept that a real shift has happened, like the increase in concern for substance abuse.