Questions: Antibiotics in Infancy A Canadian longitudinal study (1) examined whether giving antibiotics in infancy increases the likelihood that the child will be overweight later in life. The study included 616 children and found that 438 of the children had received antibiotics during the first year of life. Test to see if this provides evidence that more than 70% of Canadian children receive antibiotics during the first year of life. Show all details of the hypothesis test, including hypotheses, the standardized test statistic, the p-value, the generic conclusion using a 5% significance level, and a conclusion in context. (1) Azad MB, Bridgman SL, Becker AB, Kozyrskyj AL, "Infant antibiotic exposure and the development of childhood overweight and central adiposity". International Journal of Obesity (2014) 38, 1290-1298.

Antibiotics in Infancy
A Canadian longitudinal study (1) examined whether giving antibiotics in infancy increases the likelihood that the child will be overweight later in life. The study included 616 children and found that 438 of the children had received antibiotics during the first year of life. Test to see if this provides evidence that more than 70% of Canadian children receive antibiotics during the first year of life. Show all details of the hypothesis test, including hypotheses, the standardized test statistic, the p-value, the generic conclusion using a 5% significance level, and a conclusion in context.
(1) Azad MB, Bridgman SL, Becker AB, Kozyrskyj AL, "Infant antibiotic exposure and the development of childhood overweight and central adiposity". International Journal of Obesity (2014) 38, 1290-1298.

Transcript text: Antibiotics in Infancy A Canadian longitudinal study ${ }^{1}$ examined whether giving antibiotics in infancy increases the likelihood that the child will be overweight later in life. The study included 616 children and found that 438 of the children had received antibiotics during the first year of life. Test to see if this provides evidence that more than $70 \%$ of Canadian children receive antibiotics during the first year of life. Show all details of the hypothesis test, including hypotheses, the standardized test statistic, the $p$-value, the generic conclusion using a $5 \%$ significance level, and a conclusion in context. ${ }^{1}$ Azad MB, Bridgman SL, Becker AB, Kozyrskyj AL, "Infant antibiotic exposure and the development of childhood overweight and central adiposity". International Journal of Obesity (2014) 38, 1290-1298.

Solution

Solution Steps

Step 1: State the Hypotheses

We want to test whether more than 70% of Canadian children receive antibiotics during the first year of life. Thus, we set up the hypotheses as follows:

Null Hypothesis ($H_0$): $p \leq 0.70$
Alternative Hypothesis ($H_a$): $p > 0.70$

Step 2: Calculate the Sample Proportion

The sample proportion ($\hat{p}$) is calculated as:

\[ \hat{p} = \frac{x}{n} = \frac{438}{616} \approx 0.7110 \]

Step 3: Calculate the Test Statistic

The test statistic ($Z$) for the sample proportion is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the values:

\[ Z = \frac{0.7110 - 0.70}{\sqrt{\frac{0.70(1 - 0.70)}{616}}} \approx 0.5979 \]

Step 4: Determine the P-value

The P-value associated with the test statistic $Z = 0.5979$ is calculated to be:

\[ \text{P-value} \approx 0.275 \]

Step 5: Identify the Critical Region

For a significance level of $\alpha = 0.05$ in a one-tailed test, the critical value for $Z$ is:

\[ Z_{critical} \approx 1.6449 \]

Thus, the critical region is defined as:

\[ Z > 1.6449 \]

Step 6: Make a Decision

Since the calculated test statistic $Z = 0.5979$ does not fall into the critical region, we fail to reject the null hypothesis.

Step 7: Conclusion

Based on the results, we conclude that there is not sufficient evidence to suggest that more than 70% of Canadian children receive antibiotics during the first year of life.

Final Answer

$\boxed{\text{There is not sufficient evidence to suggest that more than 70% of Canadian children receive antibiotics during the first year of life.}}$

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