Question

Calculate the lower confidence limit (LCL) and upper confidence limit (UCL) of the mean for each of the following. a. xˉ=395,n=403,σ=40, and α=0.01 b. xˉ=245,n=84,σ2=64, and α=0.05 Click the icon to view the standard normal table of the cumulative distribution function. a. LCL= (Round to two decimal places as needed.) UCL= (Round to two decimal places as needed.) b. LCL= (Round to two decimal places as needed.) UCL= (Round to two decimal places as needed.)

Answer 1

To calculate the LCL and UCL for each scenario, use the formula LCL =x - (Z(α/2))(σ/√n) and UCL =x + (Z(α/2))(σ/√n), where Z(α/2) is the critical Z-value found using standard Z-tables.

The write answer is

a. LCL = 392.86, UCL = 397.14

b. LCL = 238.84, UCL = 251.16

Step-by-step explanation:

To calculate the lower confidence limit (LCL) and the upper confidence limit (UCL) for the mean, you need the sample mean (x the sample size (n), the population standard deviation (σ), and the alpha level (α).

For a normal distribution, the confidence level (CL) formula is CL = 1 - α. We will use the standard normal Z-distribution to find the critical Z-value (Zα/2) because we assume the population standard deviation is known. The formula for the confidence limits is then:

LCL =x - (Zα/2)(σ/√n)
UCL = x+ (Zα/2)(σ/√n)

For calculation a):

Given x=395, n=403, σ=40, and α=0.01,
we find Z0.005 since α/2 = 0.005, which corresponds to a Z-value of approximately 2.576 (using standard Z-tables). Then calculate the LCL and UCL.

For calculation b):

Given x=245, n=84, σ^2=64, so σ=8, and α=0.05,
we find Z0.025 since α/2 = 0.025, which corresponds to a Z-value of approximately 1.96. Then calculate the LCL and UCL.

The write answer is

a. LCL = 392.86, UCL = 397.14

b. LCL = 238.84, UCL = 251.16