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Chapter 8: Problem 26

According to a Gallup poll, \(11.55 \%\) of American adults have diabetes. Suppose a researcher wonders if the diabetes rate in her area is higher than the national rate. She surveys 150 adults in her area and finds that 21 of them have diabetes. a. If the region had the same rate of diabetes as the rest of the country, how many would we expect have diabetes? b. Suppose you are testing the hypothesis that the diabetes rate in this area differs from the national rate, using a \(0.05\) significance level. Choose the correct figure and interpret the p-value.

Short Answer

Expert verified
The expected number of people having diabetes, given the same rate as the rest of the country, would be about 17. To determine whether the diabetes rate in that area differs from the national rate, a hypothesis test needs to be performed. This involves calculating a z-score and comparing the p-value retrieved from the z-table with the significance level.

Step by step solution

01

Find the Expected Value

The expected number of diabetes cases can be found by multiplying the Gallup poll's rate by the sample size of 150. Mathematically, this is expressed as:\(Expected Value = Sample Size * Rate= 150 * 0.1155 = 17.325 \). The fractional number should be rounded to nearest integer as you cannot have a fraction of a person. Hence, the expected number of diabetes cases would be 17.
02

Perform the Hypothesis Test

In order to test whether the diabetes rate in the researcher's area is different than the national rate, a hypothesis test must be performed. Before conducting the test, the null and alternative hypotheses must be established:Null Hypothesis (\(H_0\)): The diabetes rate in the researcher's area is the same as the national rate.Alternative Hypothesis (\(H_a\)): The diabetes rate in the researcher's area is different than the national rate. After setting up the hypotheses, calculate the z-score using the formula: \(Z= \frac{(SampleRate - PopulationRate)}{\sqrt{\frac{PopulationRate*(1-PopulationRate)}{SampleSize}}} \)For this case: \(Z= \frac{(21/150 - 0.1155)}{\sqrt{\frac{0.1155*(1-0.1155)}{150}}} \)Once the z-score is calculated, find the p-value using a z-table. If the p-value is less than the significance level (in this case, 0.05), reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value Calculation
The expected value in statistics is a fundamental concept, representing the average outcome if an experiment were repeated many times. When applied to real-world scenarios, such as estimating the number of people with a particular condition, it provides us with a vital benchmark. For the given exercise, the expected value is the projected number of individuals with diabetes in the researcher's area, based on the national diabetes rate. To compute it, we use the formula:
\( Expected Value = Sample Size \times Rate \)
With the sample size being 150 adults and the rate at 11.55%, the expected number would mathematically be 17.325, which we round to 17 to reflect the whole number of individuals.
Null and Alternative Hypothesis
In hypothesis testing, we work with two competing hypotheses: the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_a \)). The null hypothesis represents the status quo, suggesting no effect or no difference. In contrast, the alternative hypothesis indicates there is an effect or a difference present. For our exercise, the null hypothesis posits that the diabetes rate in the area is equal to the national rate. In contrast, the alternative hypothesis suggests that the rate in the area is not the same as the national rate. Establishing these hypotheses is crucial before proceeding with any statistical testing.
Statistical Significance
Statistical significance tells us whether the observed data is sufficiently unusual compared to a null hypothesis. It helps researchers decide if they can reject the null hypothesis, potentially leading to accepting the alternative hypothesis. The significance level (\( \alpha \)), often set at 0.05, acts as a threshold. If the test statistic, such as a p-value, is below this level, the result is considered statistically significant. This concept informs us whether or not the findings provide strong evidence against the null hypothesis in the context of the given problem.
P-value Interpretation
The p-value is a probabilistic measure indicating how extreme the data is, assuming the null hypothesis is true. A smaller p-value suggests that the observed data is less likely under the null hypothesis. Interpreting p-values in terms of the \( \alpha \) level provides a clear decision rule: If our p-value is less than \( \alpha \), we reject the null hypothesis, as it indicates that the observed results are unusual under the assumption that \( H_0 \) is correct. It is crucial for students to understand that a p-value is not the probability that the null hypothesis is true or false but a measure of the evidence against it.
Z-score Calculation
A z-score is a standard score that indicates how many standard deviations an element is from the mean. It is essential in hypothesis testing for comparing the observed results with what is expected under the null hypothesis. The formula for z-score calculation in the context of a proportion test is:
\( Z = \frac{(SampleRate - PopulationRate)}{\sqrt{{\frac{PopulationRate \times (1 - PopulationRate)}{SampleSize}}}} \)
In our scenario, the SampleRate is the observed proportion of diabetes cases in the sample, while the PopulationRate is the national diabetes rate. The z-score tells us the direction and magnitude of the deviation from the expected value. Students need to be comfortable with this calculation as it's a critical component of hypothesis testing for determining the statistical significance of the results.

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Most popular questions from this chapter

According to one source, \(50 \%\) of plane crashes are due at least in part to pilot error (http://www.planecrashinfo .com). Suppose that in a random sample of 100 separate airplane accidents, 62 of them were due to pilot error (at least in part.) a. Test the null hypothesis that the proportion of airplane accidents due to pilot error is not \(0.50 .\) Use a significance level of \(0.05\). b. Choose the correct interpretation: i. The percentage of plane crashes due to pilot error is not significantly different from \(50 \%\). ii. The percentage of plane crashes due to pilot error is significantly different from \(50 \%\).

Refer to Exercise \(8.97 .\) Suppose 14 out of 20 voters in Pennsylvania report having voted for an independent candidate. The null hypothesis is that the population proportion is \(0.50 .\) What value of the test statistic should you report?

According to the Bureau of Labor Statistics, \(10.1 \%\) of Americans are self- employed. A researcher wants to determine if the self-employment rate in a certain area is different. She takes a random sample of 500 working residents from the area and finds that 62 are self-employed. a. Test the hypothesis that the proportion of self-employed workers in this area is different from \(10.1 \%\). Use a \(0.05\) significance level. b. After conducting the hypothesis test, a further question one might ask. "What proportion of workers in this area are self-employed?" Use the sample data to find a \(95 \%\) confidence interval for the proportion of workers who are self-employed in the area from which the sample was drawn. How does this confidence interval support the hypothesis test conclusion?

Choosing a Test and Giving the Hypotheses Give the null and alternative hypotheses for each test, and state whether a one-proportion z-test or a two- proportion z-test would be appropriate. a. You test a person to see whether he can tell tap water from bottled water. You give him 20 sips selected randomly (half from tap water and half from bottled water) and record the proportion he gets correct to test the hypothesis. b. You test a random sample of students at your college who stand on one foot with their eyes closed and determine who can stand for at least 10 seconds, comparing athletes and nonathletes.

Biased Coin? A study is done to see whether a coin is biased. The alternative hypothesis used is two-sided, and the obtained z-value is 2. Assuming that the sample size is sufficiently large and that the other conditions are also satisfied, use the Empirical Rule to approximate the p-value.
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