birth weight of infants: a health care professional wishes to estimate the birth weights of infants. the researcher desires to be 99% confident that the true mean is within 2.5 ounces of the sample mean. the standard deviation of infant birth wieghts is known to be 7 ounces. show work

a.based on the information provided above, how large a sample must be obtained

b. if the resercher wanted to be 90% confident, how large should the sample size be

c. for which situation- a or b was the sample size, n, larger. explain why the calculated sample size was larger

To determine the sample size needed for estimating birth weights of infants, we can use the formula for sample size calculation with a known standard deviation. The formula is as follows:

n = [(Z * σ) / E]^2

Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (in this case, 99% or 90%)
σ = standard deviation
E = margin of error (half of the desired interval width)

a. For 99% confidence level:
Z-score for a 99% confidence level is approximately 2.576 (obtained from a standard normal distribution table).
E = 2.5 ounces / 2 = 1.25 ounces (half of the desired interval width)
σ = 7 ounces

Substituting the values into the formula:
n = [(2.576 * 7) / 1.25]^2
n ≈ 46.62

Therefore, the researcher should obtain a sample size of at least 47 infants to estimate birth weights, ensuring a 99% confidence level and a margin of error of 2.5 ounces.

b. For 90% confidence level:
Z-score for a 90% confidence level is approximately 1.645 (obtained from a standard normal distribution table).
E = 2.5 ounces / 2 = 1.25 ounces (half of the desired interval width)
σ = 7 ounces

Substituting the values into the formula:
n = [(1.645 * 7) / 1.25]^2
n ≈ 22.13

Therefore, for a 90% confidence level and a margin of error of 2.5 ounces, the researcher should obtain a sample size of at least 23 infants.

c. The sample size was larger for situation (a) because a higher confidence level requires a larger sample size. When the confidence level increases, the Z-score also increases, resulting in a larger required sample size. In this case, for a 99% confidence level (situation a), the estimated sample size was 47, whereas for a 90% confidence level (situation b), the estimated sample size was 23.