Questions: Babies: According to a recent report, a sample of 300 one-year-old baby boys in the United States had a mean weight of 25.5 pounds. Assume the population standard deviation is σ=5.2 pounds. Part 1 of 3 (a) Construct a 90% confidence interval for the mean weight of all one-year-old baby boys in the United States. Round the answer to at least one decimal place. A 90% confidence interval for the mean weight in pounds of all one-year-old baby boys in the United States is <μ<.

Solution Steps

We are given the following information about a sample of one-year-old baby boys in the United States:

• Sample size (\(n\)) = 300
• Sample mean (\(\bar{x}\)) = 25.5 pounds
• Population standard deviation (\(\sigma\)) = 5.2 pounds
• Confidence level = 90%

For a 90% confidence level, the significance level (\(\alpha\)) is: \[ \alpha = 1 - 0.90 = 0.10 \] Since this is a two-tailed test, we divide \(\alpha\) by 2: \[ \alpha/2 = 0.05 \] The Z-score corresponding to a 90% confidence level is approximately \(z \approx 1.645\).

The margin of error (ME) is calculated using the formula: \[ ME = z \cdot \frac{\sigma}{\sqrt{n}} \] Substituting the values: \[ ME = 1.645 \cdot \frac{5.2}{\sqrt{300}} \approx 1.645 \cdot 0.300 \approx 0.4935 \] Rounding to one decimal place, we find: \[ ME \approx 1.6 \]

The confidence interval is given by: \[ \bar{x} \pm ME \] Calculating the lower and upper bounds: \[ \text{Lower bound} = 25.5 - 1.6 = 23.9 \] \[ \text{Upper bound} = 25.5 + 1.6 = 27.1 \]

Thus, the 90% confidence interval for the mean weight of all one-year-old baby boys in the United States is: \[ (23.9, 27.1) \]

Final Answer