construct the confidence interval for the population mean c=0.98,x=5.6, s=0.6, and n=45
To construct a confidence interval for the population mean, you can use the formula:
CI = x ± Z * (s / √n)
Where:
CI is the confidence interval
x is the sample mean
Z is the z-score corresponding to the desired confidence level
s is the sample standard deviation
n is the sample size
Given: c = 0.98, x = 5.6, s = 0.6, and n = 45
Step 1: Determine the Z-score
Since the confidence level is 0.98, the corresponding alpha value is 1 - 0.98 = 0.02 (since the confidence level is usually expressed as 1 - alpha). We need to find the Z-score that corresponds to the alpha divided by 2, since we want to include both tails in the confidence interval.
Using a Z-table or a statistical calculator, we find that the Z-score for a 0.02/2 = 0.01 significance level (two-tailed) is approximately 2.58.
Step 2: Calculate the confidence interval
Using the formula:
CI = x ± Z * (s / √n)
Substituting the given values:
CI = 5.6 ± 2.58 * (0.6 / √45)
Calculating the value inside the parentheses:
s / √n = 0.6 / √45 ≈ 0.089
Substituting this value into the formula:
CI = 5.6 ± 2.58 * 0.089
Calculating the confidence interval:
Lower Bound = 5.6 - (2.58 * 0.089) ≈ 5.34
Upper Bound = 5.6 + (2.58 * 0.089) ≈ 5.86
Therefore, the confidence interval for the population mean is approximately (5.34, 5.86) at a confidence level of 0.98.