Using the central limit theorem, construct a 95% confidence interval for the average total cry time for infants in the control group and infants in the intervention group. For this question only, assume that the standard deviation of cry time within each group is known and is equal to 22 seconds.
95% = mean ± 1.96 SEm
SEm = SD/√n
What is your mean and n for each group?
To construct a 95% confidence interval for the average total cry time for infants in the control group and intervention group, you can follow these steps using the central limit theorem:
1. Determine the sample size for each group. Let's say the control group has a sample size of n1 and the intervention group has a sample size of n2.
2. Calculate the mean cry time for each group. Let's denote the mean cry time for the control group as x̄1 and for the intervention group as x̄2.
3. Since we assume the standard deviation within each group is known (denoted as σ), we can calculate the standard error for each group using the formula: standard error = σ / √(sample size). For the control group, the standard error would be σ1 = σ / √(n1), and for the intervention group, the standard error would be σ2 = σ / √(n2).
4. Determine the z-score for a 95% confidence interval, which corresponds to a 1 - (1-0.95)/2 = 0.975 cumulative probability. You can find this value in a standard normal distribution table or use a statistical software, and it is approximately 1.96.
5. Calculate the margin of error for each group, which is the z-score multiplied by the corresponding standard error. For the control group, the margin of error would be ME1 = 1.96 * σ1, and for the intervention group, the margin of error would be ME2 = 1.96 * σ2.
6. Finally, construct the confidence interval for each group by adding and subtracting the margin of error from the corresponding sample mean. For the control group, the confidence interval would be x̄1 ± ME1, and for the intervention group, the confidence interval would be x̄2 ± ME2.
Keep in mind that these calculations assume that the sample sizes are large enough for the central limit theorem to hold. If the sample sizes are small, you may need to use a different method, such as a t-distribution.
By following these steps, you should be able to construct a 95% confidence interval for the average total cry time for infants in the control group and intervention group using the central limit theorem.