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Chapter 20: Problem 33

A new vaccine was recently tested to see if it could prevent the painful and recurrent ear infections that many infants suffer from. The Lancet, a medical journal, reported a study in which babies about a year old were randomly divided into two groups. One group received vaccinations; the other did not. During the following year, only 333 of 2455 vaccinated children had ear infections, compared to 499 of 2452 unvaccinated children in the control group. a. Are the conditions for inference satisfied? b. Find a \(95 \%\) confidence interval for the difference in rates of ear infection. c. Use your confidence interval to explain whether you think the vaccine is effective.

Short Answer

Expert verified
The rate of ear infection in vaccinated children is between \(3.34 \%\) and \(10.18 \%\) lesser than in unvaccinated children with \(95 \%\) confidence, indicating that the vaccine may effectively reduce ear infections in infants.

Step by step solution

01

Check Conditions For Inference

There are two main conditions for making inferences about a population proportion. The first is random sampling, which is satisfied here as the babies were randomly divided into two groups. The second is that the sample size should be sufficiently large, which we can check by ensuring that \(np \geq 10\) and \(n(1-p) \geq 10\). Plugging in the numbers we get for vaccinated group \(2455 * (333/2455) = 333 \geq 10\) and \(2455 * (1 - 333/2455) = 2122 \geq 10\). For unvaccinated group \(2452 * (499/2452) = 499 \geq 10\) and \(2452 * (1 - 499/2452) = 1953 \geq 10\). So, the condition is satisfied.
02

Calculate Point Estimate and Standard Error

The point estimate for the difference in proportions is \(p_1 - p_2 = (333/2455) - (499/2452) = -0.0676\). The standard error for this estimate is \(\sqrt{[p_1(1-p_1)/n_1] + [p_2(1-p_2)/n_2]} = \sqrt{[(333/2455)(1-333/2455)/2455] + [(499/2452)(1-499/2452)/2452]} = 0.0174\).
03

Calculate Confidence Interval

A \(95 \%\) confidence interval for the difference in proportions is given by the formula \((p_1-p_2) \pm z*SE\), where \(z = 1.96\) for \(95 \%\) confidence. Plug the values from previous calculations to get \(-0.0676 \pm 1.96*0.0174\), which gives \(-0.1018, -0.0334\) as the confidence interval.
04

Interpret Results

With \(95 \%\) confidence, the proportion of vaccinated children getting ear infections is between \(3.34 \%\) and \(10.18 \%\) less than the proportion of unvaccinated children getting ear infections.

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Key Concepts

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

Vaccination Effectiveness
Understanding the effectiveness of a vaccine is key in public health. Here, we are exploring whether a new vaccine can lower the incidence of ear infections in infants.
We can gauge the effectiveness of the vaccine by observing the difference in infection rates between vaccinated and unvaccinated groups.
This examination was conducted using a statistical method called 'difference in proportions'. A vaccine is considered effective if the vaccinated group displays a significantly lower proportion of infections compared to the unvaccinated group. In our study, only 333 out of 2455 vaccinated children got ear infections, whereas 499 out of 2452 unvaccinated children experienced them.
When we convert these into proportions, this gives 13.6% for the vaccinated group and 20.3% for the unvaccinated group. This suggests that the vaccine could lower the risk of infections significantly. Effectiveness isn't just about percentages, though. It also involves statistical tools, such as confidence intervals, to determine how reliably these results suggest a real difference that isn't just due to chance.
Statistical Inference
Statistical inference allows us to make conclusions about a population based on a sample. Here, inference helps us determine if the new vaccine effectively reduces ear infection rates.
For reliable statistical inference, certain conditions must be met, such as random sampling and adequate sample size. In this exercise, babies were randomly divided into two groups鈥攖hose vaccinated and those not vaccinated.
This randomization satisfies one condition for inference, minimizing the risk of bias. The second condition concerns the sample size, which must be large enough to make valid predictions.
In our case, both groups exceeded the size threshold, meaning that statistical tests are less likely to give inaccurate results. Therefore, we can use statistical inference to confidently assert that the observed differences in infection rates are not accidental.
Difference in Proportions
The difference in proportions is a measure used to examine the effectiveness of treatment or intervention between two groups.
In this study, we want to find out the difference in proportions between the vaccinated and unvaccinated groups. This difference can tell us how effective the vaccination is in preventing infections.First, we calculate the point estimate of the difference. The formula is simple: \[ p_1 - p_2 \]where \( p_1 \) and \( p_2 \) are the infection rates for the vaccinated and unvaccinated groups, respectively.
For this study, the point estimate is -0.0676, indicating a lower infection rate in the vaccinated group. To understand how precisely this point estimate reflects the true difference, we construct a confidence interval. A 95% confidence interval, for example, would show us a range in which we are 95% certain the true difference in proportions lies. This methodical approach allows us to back up claims about vaccine effectiveness with statistical evidence, thus assuring us that the results are robust and not merely due to random chance. In this scenario, the negative interval indicates the vaccinated group indeed has fewer infections.

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Most popular questions from this chapter

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One month before the election, a poll of 630 randomly selected voters showed \(54 \%\) planning to vote for a certain candidate. A week later, it became known that he had had an extramarital affair, and a new poll showed only \(51 \%\) of 1010 voters supporting him. Do these results indicate a decrease in voter support for his candidacy? a. Test an appropriate hypothesis and state your conclusion. b. If you concluded there was a difference, estimate that difference with a confidence interval and interpret your interval in context.

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