Final answer:
The distribution of X, the average number of miles a car's tire will function before needing replacement, follows a normal distribution. To solve this question, we need to find the probability that a random sample of 47 tires will have an average lifespan less than a certain threshold. This can be done by converting the sample mean into a Z-score and using the standard normal distribution table or calculator to find the probability.
Step-by-step explanation:
The distribution of X, the average number of miles a car's tire will function before needing replacement, follows a normal distribution. Given that the average number of miles is 65 (in thousands) and the standard deviation is 20, we can represent this as X ~ N(65, 20).
To solve this question, we need to find the probability that a random sample of 47 tires will have an average lifespan less than a certain threshold. This can be done by converting the sample mean into a Z-score and using the standard normal distribution table or calculator to find the probability.
For example, if we want to find the probability that the average lifespan is less than 60, we can calculate the Z-score as (60 - 65) / (20 / sqrt(47)). We can then use the Z-score to find the corresponding probability from the standard normal distribution.