You invent a new system to make excellent racing tires for the Indianapolis 500/ Your tires are good for an average of 105 miles, with a standard deviation of 7.5 miles. What is the probability that one of your tires can go for at least 44 laps? What is the probability that one of your tires will fail before the end of 36 laps?
How far is a lap?
One lap is 2.5 miles.
Sorry, meant to include that before.
so 44 laps = 44(2.5) or 110 miles
36 laps = 90 miles
enter your values in
http://davidmlane.com/hyperstat/z_table.html
enter 110 for 44 laps in "above" ---> .2525
enter 90 and click on "below" ------> .02275
To solve these questions, we will use the concept of z-scores and the standard normal distribution. The z-score allows us to compare a specific value to the mean in terms of standard deviations.
To find the probability that one of your tires can go for at least 44 laps, we need to find the area under the normal distribution curve to the right of 44 laps.
Step 1: Calculate the z-score
The formula for z-score is (x - μ) / σ, where x is the given score, μ is the mean, and σ is the standard deviation.
In this case, we want to find the z-score for 44 laps: z = (44 - 105) / 7.5 = -7.47.
Step 2: Find the probability using the z-table
The z-table provides the probability corresponding to a given z-score. Since our z-score is negative, we need to find the area to the right of the z-score (1 minus the area to the left).
Looking up the z-score of -7.47 in the z-table, we find that the area to the left of it is close to 0 (approximately 0.0000000). Therefore, the area to the right is 1 - 0.0000000 ≈ 1.
Hence, the probability that one of your tires can go for at least 44 laps is approximately 1 (or 100%).
Now let's move on to the second question.
To find the probability that one of your tires will fail before the end of 36 laps, we need to find the area under the normal distribution curve to the left of 36 laps.
Step 1: Calculate the z-score
Using the same formula, z = (36 - 105) / 7.5 = -10.8.
Step 2: Find the probability using the z-table
Since the z-score is negative, we need to find the area to the left. Looking up the z-score of -10.8 in the z-table, we find that the area to the left is close to 0 (approximately 0.0000000).
So, the probability that one of your tires will fail before the end of 36 laps is approximately 0 (or 0%).
Please keep in mind that these calculations assume a normal distribution and may not perfectly represent the actual performance of your tires. Also, it's always recommended to conduct experiments or gather more data to have a better understanding of the tire performance.