b. The top-selling Amar tire is rated 70,000 KMs, which means nothing. In fact, the distance the tires can run until they wear out is a normally distributed random variable with a mean of 82,000 KMs and a standard deviation of 6,400 KMs.
What is the probability that a tire wears out before 70,000 KMs?
What is the probability that a tire lasts more than 100,000 KMs
You can play around with Z table stuff at
davidmlane.com/hyperstat/z_table.html
To find the probability that a tire wears out before 70,000 KMs, you can use the concept of the standard normal distribution. Here's how you can calculate it:
Step 1: Standardize the value of 70,000 KMs using the formula:
Standardized Value = (X - Mean) / Standard Deviation
In this case, X represents the value 70,000 KMs, Mean represents the mean of 82,000 KMs, and Standard Deviation represents the standard deviation of 6,400 KMs.
Standardized Value = (70,000 - 82,000) / 6,400
Step 2: Once you have the standardized value, you can use a standard normal distribution table (also known as Z-table) to look up the corresponding probability. The Z-table gives you the probability of a standard normal random variable being less than or equal to a given value.
Let's assume the standardized value from Step 1 is -1.875 (rounded to three decimal places). Looking it up in the Z-table, you can find the probability.
The probability of a tire wearing out before 70,000 KMs is equal to the probability of a standardized value less than or equal to -1.875. From the Z-table, this probability is approximately 0.0304 or 3.04%.
Now, to find the probability that a tire lasts more than 100,000 KMs, follow a similar process:
Step 1: Standardize the value of 100,000 KMs using the formula:
Standardized Value = (X - Mean) / Standard Deviation
Here, X is 100,000 KMs, Mean is 82,000 KMs, and Standard Deviation is 6,400 KMs.
Standardized Value = (100,000 - 82,000) / 6,400
Step 2: Again, use the Z-table to find the probability of a standardized value greater than the value obtained from Step 1 (in this case, suppose it is 2.812).
The probability of a tire lasting more than 100,000 KMs is equal to the probability of a standardized value greater than 2.812. From the Z-table, this probability is approximately 0.0026 or 0.26%.
So, to summarize the results:
- The probability that a tire wears out before 70,000 KMs is approximately 0.0304 or 3.04%.
- The probability that a tire lasts more than 100,000 KMs is approximately 0.0026 or 0.26%.