Suppose a car rental company wants to estimate the average number of kilometres travelled per day by each of its cars rented in Adelaide. A random sample of 20 cars rented in Adelaide reveals that the sample means travel distance per day is 85.5 km per day, with a population standard deviation of 19.3 km. Assuming that the population is normally distributed, calculate the 99% confidence interval to estimate μ ( population mean ) .

She uses the data to construct a 99% confidence interval to estimate the average number of kilometres travelled per day by each car rented in Adelaide. She assumes that the average number of days per rental for the population is normally distributed.

Calculate the Lower limit

Question

Suppose a car rental company wants to estimate the average number of kilometres travelled per day by each of its cars rented in Adelaide. A random sample of 20 cars rented in Adelaide reveals that the sample means travel distance per day is 85.5 km per day, with a population standard deviation of 19.3 km. Assuming that the population is normally distributed, calculate the 99% confidence interval to estimate μ ( population mean ) .

She uses the data to construct a 99% confidence interval to estimate the average number of kilometres travelled per day by each car rented in Adelaide. She assumes that the average number of days per rental for the population is normally distributed.

Calculate the Lower  limit

BlackTom AI · Accepted Answer

We start by identifying the goal: construct a 99% confidence interval for the population mean μ given that the population standard deviation is known. Since we know σ = 19.3 km and the sample size is n = 20, we use the z-distribution for the margin of error.

Step 1: Determine the critical value for a 99% two-sided confidence interval. The standard normal z-value that leaves 0.005 in each tail is z* ≈ 2.576.

Step 2: Compute the standard error of the mean. with known σ, SE = σ / sqrt(n) = 19.3 / sqrt(20). Numerically, sqrt(20) ≈ 4.472, so SE ≈ 19.3 / 4.472 ≈ 4.316.

Step 3: Calculate the margin of error. ME = z* × SE ≈ 2.576 × 4.316. Multiplying gives ME ≈ 11.12.

Step 4: Build the confidence interval around the sample mean. The sample mean is x̄ = 85.5. The lower limit is: x̄ − ME ≈ 85.5 − 11.12 ≈ 74.38.

Thus, the computed lower limit of the 99% confidence interval for μ is approximately 74.38 km per day. Since the problem asks specifically for the lower limit, we report this value. Note that this approach relies on the population standard deviation being known and the population distribution being normal, which justifies using the z-distribution rather than the t-distribution. If σ were unknown, we would instead use the t-distribution with df = n − 1, yielding a different margin of error. Also, the normality assumption is what legitimizes applying these standard confidence interval formulas to the mean of a normally distributed population.

BSTA011 Week 7 Practice Quiz

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