91Ó°ÊÓ

A study was carried out to compare two different methods, injection and nasal spray, for administering flu vaccine to children under the age of 5 . All 8000 children in the study were given both an injection and a spray. However, the vaccine given to 4000 of the children actually contained just saltwater, and the spray given to the other 4000 children also contained just saltwater. At the end of the flu season, it was determined that \(3.9 \%\) of the children who received the real vaccine via nasal spray contracted the flu, whereas \(8.6 \%\) of the 4000 children receiving the real vaccine via injection contracted the flu. a. Why do you think each child received both an injection and a spray? b. Does one method for delivering the vaccine appear to be superior to the other? Test the appropriate hypotheses. [Note: The study was described in the article "Spray Flu Vaccine May Work Better Than Injections for Tots," San Luis Obispo Tribune, May 2, 2006.]

Step by step solution

01

Understand the Setup

Each child in the study received both a nasal spray and an injection to administer the flu vaccine. However, for the control groups, one method was a placebo: half of the children received a placebo nasal spray, the other half received a placebo injection. This controls for the action of receiving a treatment influencing results (placebo effect).

02

Analyze Question A

The main reason each child received both an injection and a spray, with one always being a placebo, is to ensure that each child experiences both methods but only receives the effective vaccine in one form. This helps to eliminate the potential placebo effect and allows the study to effectively measure which method is more effective when controlling for external treatment influences.

03

Define Hypotheses for Question B

To determine which method is superior, we test the hypotheses: - Null Hypothesis ( H_0 ): There is no difference in effectiveness between the nasal spray and injection methods. - Alternative Hypothesis ( H_a ): There is a difference in effectiveness, specifically that one method is superior.

04

Determine Statistical Test

Since we are comparing proportions of children contracting flu between two groups, use a two-proportion z-test. The proportion of children getting flu after real nasal spray is 3.9%, and after real injection is 8.6%.

05

Calculate Test Statistic

Calculate the test statistic using:\[z = \frac{(p_1 - p_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}\]where \(p_1 = 0.039\), \(p_2 = 0.086\), \(n_1 = n_2 = 4000\), and \(\hat{p}\) is the pooled sample proportion.

06

Calculate Pooled Proportion

Calculate the pooled proportion \(\hat{p}\):\[\hat{p} = \frac{0.039 \times 4000 + 0.086 \times 4000}{4000 + 4000} = \frac{156 + 344}{8000} = 0.0625\]

07

Compute Z-value

Substitute values into the z-formula:\[z = \frac{0.039 - 0.086}{\sqrt{0.0625(1-0.0625)(\frac{1}{4000} + \frac{1}{4000})}}\]Calculate the exact z-value.

08

Obtain the P-value

Using the calculated z-value, determine the p-value from standard normal distribution tables. If the p-value is less than the significance level (commonly 0.05), reject the null hypothesis.

09

Draw Conclusion

If the p-value is small, conclude that the nasal spray is statistically more effective than the injection. If not, there isn't significant evidence to say one is superior.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Proportion Comparison

Understanding proportion comparison is critical when analyzing situations where outcomes are measured across different groups. In the context of the flu vaccine study, we are comparing the proportion of children contracting the flu between two methods: nasal spray and injection. Proportion comparison answers how these proportions differ, indicating if one method might be more effective.

Both methods were applied to 4000 children, with the proportion of children contracting the flu calculated for each group. By comparing these proportions, researchers can draw inferences about the relative effectiveness of the vaccine methods. Proportion comparison allows us to numerically express how these methods perform in terms of reducing flu cases.

This analysis forms the backbone of hypothesis testing, a fundamental statistical technique used in deciding if there is enough evidence to say one method surpasses the other in effectiveness.

Two-proportion Z-test

The two-proportion Z-test is a statistical method used to determine if there is a significant difference between the proportions of two groups. In our study, it checks if the nasal spray or injection method results in a significantly different proportion of children contracting the flu.

Here's how it works:

• Identify the two independent groups—children given the nasal spray and those given the injection.
• Determine the success proportions for each group (those contracting the flu).
• Calculate the Z-test statistic, which uses both proportions and considers sample sizes to assess if observed differences are due to chance.

The outcome of the test aids us in making a decision regarding the null hypothesis, which assumes no difference in effectiveness between the two methods. By analyzing the Z-test results and p-value, we can draw significant conclusions about the superiority of one method over the other.

Placebo Effect

The placebo effect is a fascinating phenomenon where a patient experiences a perceived improvement in condition, solely because they believe they are receiving treatment. In our study, the placebo is the saltwater solution, which resembles the actual vaccine but has no therapeutic effect.

Incorporating a placebo helps by ensuring that any changes observed in the study (e.g., flu contraction rates) are due to the actual treatment method and not psychological factors attributed to just taking action.

By administering both a nasal spray and injection but using a placebo for one, the study’s design controls for placebo influences. This allows a more accurate comparison of how effective the actual nasal spray and injection methods truly are.

Pooled Sample Proportion

The pooled sample proportion is a critical component in the calculation of the two-proportion Z-test. It is a weighted average of the success proportions across all samples and is used when testing hypotheses concerning two sample proportions.

In our vaccine study, calculate the pooled proportion by aggregating the number of children who contracted the flu from both groups, divided by the total number of children in both groups.
For example:

• Children contracting flu after nasal spray: 156
• Children contracting flu after injection: 344
• Total children studied: 8000

Thus, the pooled proportion becomes:\[\hat{p} = \frac{156 + 344}{8000} = 0.0625\]This value is used within the Z-test formula, facilitating the calculation of the Z-statistic and ultimately leading to the p-value determination. It’s pivotal for ensuring the accurate interpretation of the statistical test, grounding our analysis in reliable math.