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Ear infections 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.

Step by step solution

01

Check Randomness and Independence

The first condition to check for inference is whether the samples are random and independent. The problem states that babies were "randomly divided into two groups," which satisfies the randomness requirement. Since each child is an independent subject, we also have independence.

02

Verify Sample Size Conditions

The next condition is to ensure both sample sizes are large enough to justify a normal approximation. Typically, we check if the expected number of successes and failures in each group are both at least 10. For the vaccinated group, the number of successes (ear infections) is 333, and the number of failures is 2122. For the unvaccinated group, the number of successes is 499, and the number of failures is 1953. All numbers exceed 10, satisfying the large sample size condition.

03

Calculate Sample Proportions

Calculate the sample proportions for each group: For the vaccinated group, the proportion of ear infections is \( \hat{p_1} = \frac{333}{2455} \approx 0.1357 \) and for the unvaccinated group, it is \( \hat{p_2} = \frac{499}{2452} \approx 0.2036 \).

04

Find Standard Error for Difference of Proportions

Use the formula for the standard error (SE) of the difference in proportions: \[ SE = \sqrt{\frac{\hat{p_1}(1-\hat{p_1})}{n_1} + \frac{\hat{p_2}(1-\hat{p_2})}{n_2}} \] where \( n_1 = 2455 \) and \( n_2 = 2452 \). Substituting the values, \( SE \approx \sqrt{\frac{0.1357(1-0.1357)}{2455} + \frac{0.2036(1-0.2036)}{2452}} \approx 0.0108 \).

05

Compute the Confidence Interval

The 95% confidence interval for the difference in proportions \( \hat{p_1} - \hat{p_2} \) is calculated using the formula: \[ (\hat{p_1} - \hat{p_2}) \pm Z^* \times SE \] where \( Z^* \) for a 95% confidence level is 1.96. So the confidence interval is \[ (0.1357 - 0.2036) \pm 1.96 \times 0.0108 = -0.0679 \pm 0.0212 \] which gives \(-0.0891\) to \(-0.0467\).

06

Interpret the Confidence Interval

The confidence interval from Step 5 does not contain zero, indicating a statistically significant difference between the two groups. Since the entire interval is negative, we conclude that the vaccinated group had a lower rate of ear infections than the unvaccinated group, suggesting that the vaccine is effective.

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Key Concepts

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

Inference Conditions

Before diving into calculating the confidence interval, it's crucial to check the inference conditions. Meeting these conditions ensures the results are reliable and valid. Here’s a simple breakdown of the key points for inference conditions:

• Randomness: The study should involve random sampling or random assignment to groups. In this case, babies were randomly divided into vaccinated and unvaccinated groups, satisfying this condition.
• Independence: Each data point (here, each baby) should be independent of the others. Given that babies are independently placed into groups, this condition holds.
• Sample Size: To apply the normal approximation, the sample should be large enough. Specifically, the number of expected successes and failures in each group should be at least 10. With 333 successes and 2122 failures in the vaccinated group, and 499 successes and 1953 failures in the unvaccinated group, both conditions are satisfied.

Difference in Proportions

Once the inference conditions are met, the next step is to understand the concept of difference in proportions. This difference helps us determine if there is a notable change between the two groups in the study.To find the difference in proportions, calculate the proportions of ear infections in each group:

• Vaccinated group: Proportion of ear infections, \( \hat{p_1} = \frac{333}{2455} \approx 0.1357\)
• Unvaccinated group: Proportion of ear infections, \( \hat{p_2} = \frac{499}{2452} \approx 0.2036\)

The difference in proportions, therefore, is \( \hat{p_1} - \hat{p_2} \). This will later be used in calculating the confidence interval.

Standard Error Calculation

After establishing the difference in proportions, calculating the standard error (SE) is the next critical step. The standard error shows how much variability is expected in the difference of proportions due to sampling.The formula for the standard error of the difference between two proportions is:\[ SE = \sqrt{\frac{\hat{p_1}(1-\hat{p_1})}{n_1} + \frac{\hat{p_2}(1-\hat{p_2})}{n_2}} \]Where:

• \( \hat{p_1}\) and \( \hat{p_2}\) are the proportions of interest for the vaccinated and unvaccinated groups respectively.
• \( n_1 \) and \( n_2 \) are the total number of participants in each group.

Using the proportions from the study:

• For the vaccinated group: \( \hat{p_1} = 0.1357\), \( n_1 = 2455\)
• For the unvaccinated group: \( \hat{p_2} = 0.2036\), \( n_2 = 2452\)

Substituting into the formula provides the SE, \( SE \approx 0.0108 \). This value is used for constructing the confidence interval to assess the effect of the vaccine.