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Chapter 10: Problem 105

Use the following information to answer the next three exercises. Neuroinvasive West Nile virus is a severe disease that affects a person鈥檚 nervous system . It is spread by the Culex species of mosquito. In the United States in 2010 there were 629 reported cases of neuroinvasive West Nile virus out of a total of 1,021 reported cases and there were 486 neuroinvasive reported cases out of a total of 712 cases reported in 2011. Is the 2011 proportion of neuroinvasive West Nile virus cases more than the 2010 proportion of neuroinvasive West Nile virus cases? Using a 1% level of significance, conduct an appropriate hypothesis test. 鈥� 鈥�2011鈥� subscript: 2011 group. 鈥� 鈥�2010鈥� subscript: 2010 group The p-value is 0.0022. At a 1% level of significance, the appropriate conclusion is a. There is sufficient evidence to conclude that the proportion of people in the United States in 2011 who contracted neuroinvasive West Nile disease is less than the proportion of people in the United States in 2010 who contracted neuroinvasive West Nile disease. b. There is insufficient evidence to conclude that the proportion of people in the United States in 2011 who contracted neuroinvasive West Nile disease is more than the proportion of people in the United States in 2010 who contracted neuroinvasive West Nile disease. c. There is insufficient evidence to conclude that the proportion of people in the United States in 2011 who contracted neuroinvasive West Nile disease is less than the proportion of people in the United States in 2010 who contracted neuroinvasive West Nile disease. d. There is sufficient evidence to conclude that the proportion of people in the United States in 2011 who contracted neuroinvasive West Nile disease is more than the proportion of people in the United States in 2010 who contracted neuroinvasive West Nile disease

Short Answer

Expert verified
d. There is sufficient evidence to conclude that the proportion in 2011 is more than in 2010.

Step by step solution

01

Define the Hypotheses

First, we define the null and alternative hypotheses. Let's denote \( p_{2011} \) as the proportion of neuroinvasive cases in 2011 and \( p_{2010} \) as the proportion in 2010.- Null Hypothesis \( (H_0): p_{2011} \leq p_{2010} \)- Alternative Hypothesis \( (H_a): p_{2011} > p_{2010} \)We are testing if the proportion in 2011 is greater than in 2010.
02

Collect Data and Proportions

Calculate the sample proportions for each year:- For 2010: \( \hat{p}_{2010} = \frac{629}{1021} \)- For 2011: \( \hat{p}_{2011} = \frac{486}{712} \)These calculations give us the sample proportion estimates for each year.
03

Level of Significance and Decision Rule

We are using a significance level \( \alpha = 0.01 \). The decision rule for a right-tailed test like this one is to reject the null hypothesis if the p-value is less than \( \alpha \). Otherwise, we do not reject the null hypothesis.
04

Evaluate P-value

The p-value given is 0.0022. Compare this to the significance level: - Since 0.0022 is less than 0.01, we reject the null hypothesis.
05

Conclusion

According to our hypothesis test, we have sufficient evidence to conclude that the proportion of neuroinvasive West Nile virus cases in 2011 is more than the proportion in 2010 at the 1% significance level.

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Key Concepts

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

Proportion
A proportion is essentially a way to express a part against a whole or a fraction of a total. In the context of this exercise, we are looking at the proportion of people who were affected by the neuroinvasive West Nile virus. To find a proportion, you divide the part by the whole. For instance, in 2010, there were 629 neuroinvasive cases out of a total of 1,021 reported cases. Therefore, the proportion for 2010 is computed as:
  • \[ \hat{p}_{2010} = \frac{629}{1021} \]

Similarly, for 2011, the proportion is:
  • \[ \hat{p}_{2011} = \frac{486}{712} \]

These proportions help us quantify and compare the occurrences year by year. Understanding proportions allows us to see the relative magnitude or rate of occurrences in a given group.
Significance Level
The significance level, often denoted by the symbol \( \alpha \), is a crucial concept in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is, in fact, true 鈥� essentially the risk of a false positive. In many scientific tests, the typical significance level is set at 0.05, but in our exercise, a more stringent level of 0.01 was chosen.
  • This implies that only a 1% chance of error is acceptable.
  • If the calculated p-value from the test is less than the chosen significance level, we reject the null hypothesis.

Choosing an appropriate significance level is important because it balances the risk of error and the need for credible evidence when making conclusions. Using a lower significance level, as in this exercise, means we demand stronger evidence before concluding that a difference exists.
Null Hypothesis
The null hypothesis, symbolized as \( H_0 \), is a statement made for statistical testing that assumes no effect or difference. It serves as the default or starting assumption for exploration. In this exercise, the null hypothesis proposed that the 2011 proportion of neuroinvasive cases was less than or equal to the 2010 proportion.
  • Formally written as: \[ H_0: p_{2011} \leq p_{2010} \]

The essence of the null hypothesis is to be a standpoint that can be disproved, validated only when not enough evidence is present to reject it. Its role is pivotal in hypothesis testing since conclusions are generally drawn as a contrast to the null hypothesis being rejected or not rejected.
Alternative Hypothesis
The alternative hypothesis, symbolized \( H_a \), is what researchers aim to support. Unlike the null hypothesis, it proposes an observed effect or difference. For the exercise at hand, the alternative hypothesis was that the 2011 proportion of neuroinvasive virus cases was greater than that of 2010:
  • \[ H_a: p_{2011} > p_{2010} \]

The alternative hypothesis is designed to capture our research interest and is what we are testing against the null hypothesis. Our end goal is to determine whether the data provide enough evidence to support this proposition over the null hypothesis. When the p-value from our data analysis is less than the significance level, it indicates enough evidence to support the alternative hypothesis, leading to a rejection of the null hypothesis.

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Most popular questions from this chapter

Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites. Is this a right-tailed, left-tailed, or two-tailed test?

Use the following information to answer the next twelve exercises. In the recent Census, three percent of the U.S. population reported being of two or more races. However, the percent varies tremendously from state to state. Suppose that two random surveys are conducted. In the first random survey, out of 1,000 North Dakotans, only nine people reported being of two or more races. In the second random survey, out of 500 Nevadans, 17 people reported being of two or more races. Conduct a hypothesis test to determine if the population percents are the same for the two states or if the percent for Nevada is statistically higher than for North Dakota. Is this a right-tailed, left-tailed, or two-tailed test? How do you know?

Use the following information to answer the next ten exercises. indicate which of the following choices best identifies the hypothesis test. a. independent group means, population standard deviations and/or variances known b. independent group means, population standard deviations and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion A recent drug survey showed an increase in use of drugs and alcohol among local high school students as compared to the national percent. Suppose that a survey of 100 local youths and 100 national youths is conducted to see if the proportion of drug and alcohol use is higher locally than nationally.

Use the following information to answer the next five exercises. A doctor wants to know if a blood pressure medication is effective. Six subjects have their blood pressures recorded. After twelve weeks on the medication, the same six subjects have their blood pressure recorded again. For this test, only systolic pressure is of concern. Test at the 1% significance level. $$ \begin{array}{|l|l|l|l|l|l|}\hline \text { Patient } & {\mathbf{A}} & {\mathbf{B}} & {\mathbf{C}} & {\mathbf{D}} & {\mathbf{E}} & {\mathbf{F}} \\\ \hline \text { Before } & {161} & {162} & {165} & {162} & {166} & {171} \\\ \hline \text { After } & {158} & {159} & {166} & {160} & {167} & {169} \\\ \hline\end{array} $$ State the null and alternative hypotheses.

Use the following information to answer the next two exercises. An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a 鈥渂iofeedback exercise program.鈥� Six subjects were randomly selected and blood pressure measurements were recorded before and after the training. The difference between blood pressures was calculated (after - before) producing the following results: \(\overline{x}_{d}=-10.2 \mathrm{sd}=8.4\) Using the data, test the hypothesis that the blood pressure has decreased after the training. If \(\alpha=0.05,\) the \(p\) -value and the conclusion are a. 0.0014; There is sufficient evidence to conclude that the blood pressure decreased after the training. b. 0.0014; There is sufficient evidence to conclude that the blood pressure increased after the training. c. 0.0155; There is sufficient evidence to conclude that the blood pressure decreased after the training. d. 0.0155; There is sufficient evidence to conclude that the blood pressure increased after the training.
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