3) A company that owns and services a fleet of cars for its sales force has found that the service lifetime of disc brake pads varies from car to car according to a Normal distribution with mean \( \mu-55,000 \) miles and standard deviation o -4500 miles. The company installs a new brand of brake pads on 8 cars. The average life on the pads on these 8 cars tums out to be 51,800 miles. What is the probability that the sample mean lifetime is 51,800 miles or less if the lifetime distribution is unchanged? (The compary takes this probability as evidence that the average lifetime of the new brand of pads is less than 55,000 miles.)

To determine the probability that the sample mean lifetime of the brake pads is **51,800 miles or less** under the assumption that the true mean lifetime remains **55,000 miles**, we can use the properties of the **sampling distribution of the sample mean**. ### Given: - **Population Mean (μ):** 55,000 miles - **Population Standard Deviation (σ):** 4,500 miles - **Sample Size (n):** 8 cars - **Sample Mean (X̄):** 51,800 miles ### Step 1: Determine the Standard Error (SE) The **standard error** of the sample mean is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{4,500}{\sqrt{8}} \approx \frac{4,500}{2.8284} \approx 1,591.55 \text{ miles} \] ### Step 2: Calculate the Z-Score The **z-score** measures how many standard errors the sample mean is away from the population mean: \[ z = \frac{X̄ - μ}{SE} = \frac{51,800 - 55,000}{1,591.55} \approx \frac{-3,200}{1,591.55} \approx -2.012 \] ### Step 3: Find the Probability Corresponding to the Z-Score Using standard normal distribution tables or a calculator: \[ P(Z \leq -2.012) \approx 0.0222 \text{ or } 2.22\% \] ### Conclusion The probability that the sample mean lifetime is **51,800 miles or less** is approximately **2.2%**. This low probability suggests that such an extreme sample mean is unlikely under the assumption that the true mean is still **55,000 miles**, providing evidence that the new brand of brake pads may indeed have a shorter average lifetime. **Answer:** Approximately a 2.2 % probability of observing such a sample mean or lower.