Final answer:
To calculate the lower and upper confidence limits of the mean, use the formula LCL = x - (z * (σ/√n)) and UCL = x + (z * (σ/√n)). For part a, LCL = 437.37 and UCL = 453.63. For part b, LCL = 319.52 and UCL = 330.48.
Step-by-step explanation:
To calculate the lower confidence limit (LCL) and upper confidence limit (UCL) of the mean, we can use the formula:
LCL = x - (z * (σ/√n))
UCL = x + (z * (σ/√n))
where x is the sample mean, σ is the population standard deviation, n is the sample size, and z is the z-score for the desired confidence level.
For part a, with x=445, n=102, σ=30, and α=0.01, the z-score for a 99% confidence level (α/2 = 0.01/2 = 0.005) is approximately 2.58. Plugging in the values, we get:
LCL = 445 - (2.58 * (30/√102)) = 437.37
UCL = 445 + (2.58 * (30/√102)) = 453.63
Therefore, the lower confidence limit (LCL) is 437.37 and the upper confidence limit (UCL) is 453.63.
For part b, with x=325, n=315, σ^2=25, and α=0.05, we need to find the z-score for a 95% confidence level (α/2 = 0.05/2 = 0.025). Using a standard normal distribution table, we find that the z-score for a 95% confidence level is approximately 1.96. Plugging in the values, we get:
LCL = 325 - (1.96 * (√25/√315)) = 319.52
UCL = 325 + (1.96 * (√25/√315)) = 330.48
Therefore, the lower confidence limit (LCL) is 319.52 and the upper confidence limit (UCL) is 330.48.