We wish to estimate the mean serum indirect bilirubin level of 4-day-old infants. The mcan for a sample of 16 infants wns found to be 5.98 mg/100 cc. Assume that bilirubin levels in 4-day-old infants are approximately normally distributcd with a standard deviation of 3.5 mg/100 cc. 6.2.2(on page 153 of 11th edition) of your textbook. For this problem, find the 0%, 95%, and 99% confidence intervals for the mean of the bilirubin levels in 4- ay-old infants! Hint: "IT AS A NOMRAL DISTRIBUTION WITH SIGMA = 3.5"
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![### Estimating Confidence Intervals for Mean Serum Indirect Bilirubin Levels in 4-Day-Old Infants
#### Problem Statement:
We wish to estimate the mean serum indirect bilirubin level of 4-day-old infants. The mean for a sample of 16 infants was found to be 5.98 mg/100 cc. Assume that bilirubin levels in 4-day-old infants are approximately normally distributed with a standard deviation of 3.5 mg/100 cc.
#### Task:
Using the data provided, calculate the 90%, 95%, and 99% confidence intervals for the mean of the bilirubin levels in 4-day-old infants.
Hint: Treat it as a normal distribution.
Given:
- Sample Mean (\( \bar{X} \)) = 5.98
- Standard Deviation (s) = 3.5
- Sample Size (n) = 16
#### Solution:
1. **90% Confidence Interval:**
- Degrees of Freedom (df) = n - 1 = 16 - 1 = 15
- \( t_{c} \) (critical t value) for 90% confidence with \( df = 15 \) is approximately 1.753.
- Formula: \( \bar{X} \pm t_{c} \frac{s}{\sqrt{n}} \):
\[
5.98 \pm 1.753 \frac{3.5}{\sqrt{16}}
\]
- Calculation:
\[
5.98 \pm 1.753 \frac{3.5}{4} \implies 5.98 \pm 1.753 \times 0.875
\]
\[
5.98 \pm 1.535 \implies (4.446, 7.514)
\]
- **90% C.I:** (4.446, 7.514)
2. **95% Confidence Interval:**
- \( t_{c} \) for 95% confidence with \( df = 15 \) is approximately 2.131.
- Formula: \( \bar{X} \pm t_{c} \frac{s}{\sqrt{n}} \):
\[
5.98 \pm 2.131 \frac{3.5}{\sqrt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bc805eb-60ed-457b-ab8e-02c516698c74%2F91885eca-dc84-492d-b8e3-43e9b2bc56e6%2Flzzi7gb_processed.jpeg&w=3840&q=75)
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