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
a. To determine the sample size required for a 99% confidence level with a 2.5 ounces margin of error, we can use the formula for the sample size:
n = (Z * σ / E)^2
Where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level (99% confidence level corresponds to a Z-score of approximately 2.576)
- σ is the known standard deviation of birth weights (7 ounces)
- E is the desired margin of error (2.5 ounces)
Plugging in the values into the formula:
n = (2.576 * 7 / 2.5)^2
n = (18.0272 / 2.5)^2
n = 7.2109^2
n = 52.0766
Therefore, the researcher must obtain a sample size of at least 53 infants to achieve a 99% confidence level with a margin of error of 2.5 ounces.
b. For a 90% confidence level, we will use the same formula with the appropriate Z-score for a 90% confidence level, which is approximately 1.645.
n = (1.645 * 7 / 2.5)^2
n = (11.4915 / 2.5)^2
n = 4.5966^2
n = 21.1290
Therefore, the researcher would need a sample size of at least 22 infants to achieve a 90% confidence level with a margin of error of 2.5 ounces.
c. The sample size for situation a (99% confidence level) is larger than situation b (90% confidence level). This is because as the desired confidence level increases, the sample size needed also increases. This relationship occurs because a higher confidence level requires a narrower margin of error, which can only be achieved by increasing the sample size. In other words, to be more confident in the estimate, more data points (infants) need to be included in the sample.