You measure 24 backpacks' weights, and find they have a mean weight of 39 ounces. Assume the population standard deviation is 5.5 ounces. Based on this, construct a 99% confidence interval for the true population mean backpack weight.
Give your answers as decimals, to two places
To construct a 99% confidence interval, we will use the formula:
CI = x̄ ± Z * (σ/√n)
Where:
CI = Confidence Interval
x̄ = Sample mean
Z = Z-score corresponding to the desired confidence level
σ = Population standard deviation
n = Sample size
We are given:
Sample mean (x̄) = 39 ounces
Population standard deviation (σ) = 5.5 ounces
Sample size (n) = 24
The Z-score corresponding to a 99% confidence level can be found using a Z-table or calculator. In this case, the Z-score is approximately 2.58.
Plugging the values into the formula:
CI = 39 ± 2.58 * (5.5/√24)
Calculating the standard error (SE):
SE = σ/√n = 5.5/√24 ≈ 1.12
CI = 39 ± 2.58 * 1.12
Calculating the upper and lower bounds of the confidence interval:
Upper bound = 39 + 2.58 * 1.12 ≈ 41.03
Lower bound = 39 - 2.58 * 1.12 ≈ 36.97
Therefore, the 99% confidence interval for the true population mean backpack weight is approximately 36.97 to 41.03 ounces.