According to a study by the Centers for Disease Control, the national mean hospital stay after childbirth is 2.1 days. Reviewing records at her own hospital, a hospital administrator calculates that the mean hospital stay for a random sample of 81 women after childbirth is 2.3 days with a standard deviation of 1.2 days.
The 95% confidence interval is ( days, days) Show two decimal places in your responses.
How do I do this?
95% = mean ± 1.96 SEm
SEm = SD/√n
To calculate the 95% confidence interval for the mean hospital stay after childbirth, you can use the formula:
CI = X̄ ± (Z * (σ / √n))
Where:
- CI represents the confidence interval
- X̄ represents the sample mean
- Z represents the critical value (which can be found using a table or calculator) corresponding to the desired confidence level
- σ represents the population standard deviation
- n represents the sample size
In this case, the sample mean is 2.3 days and the standard deviation is 1.2 days. Since you are given a sample size of 81, you need to find the critical value corresponding to a 95% confidence level.
To find the critical value, you can use a standard normal distribution table or calculator. For a 95% confidence level, the critical value (Z) is approximately 1.96.
Now let's substitute the values into the formula:
CI = 2.3 ± (1.96 * (1.2 / √81))
Calculating this, we get:
CI = 2.3 ± (1.96 * (1.2 / 9))
Simplifying further, we get:
CI = 2.3 ± (1.96 * 0.133)
Now, calculate the upper and lower limits of the confidence interval:
Upper limit = 2.3 + (1.96 * 0.133)
Lower limit = 2.3 - (1.96 * 0.133)
Rounding to two decimal places, the 95% confidence interval for the mean hospital stay after childbirth is:
(2.03, 2.57) days