the life spans of three randomly selected tires are 31,400 miles, 40,600 miles and 36,000 miles. Using the empirical rule, find the percentile that corresponds to each life span.

life's span miles what's percentile

36,850
41,150
39,000

To find the percentile that corresponds to each life span using the empirical rule, we first need to calculate the z-scores for each tire's life span.

The z-score measures how many standard deviations an individual value is from the mean. It can be calculated using the formula:

z = (x - μ) / σ

where:
- x is the individual value (life span of the tire)
- μ is the mean of the data set
- σ is the standard deviation of the data set

Let's calculate the z-scores for each tire's life span:

1. Tire 1: Life span = 31,400 miles
- Find the mean (μ) and standard deviation (σ) of the data set.
- In this case, we don't have the entire data set, so let's assume it follows a normal distribution. We can estimate the mean and standard deviation using the given data points.
- μ ≈ (31,400 + 40,600 + 36,000) / 3 ≈ 36,000 miles (estimated)
- Calculate the sample standard deviation:
- s = sqrt(((31,400 - 36,000)^2 + (40,600 - 36,000)^2 + (36,000 - 36,000)^2) / 2) ≈ 4,080 miles (estimated)
- Calculate the z-score for Tire 1:
- z = (31,400 - 36,000) / 4,080 ≈ -1.13

2. Tire 2: Life span = 40,600 miles
- Since we already estimated the mean (μ) and standard deviation (σ), we can use the same values.
- Calculate the z-score for Tire 2:
- z = (40,600 - 36,000) / 4,080 ≈ 1.13

3. Tire 3: Life span = 36,000 miles
- Calculate the z-score for Tire 3 using the estimated mean and standard deviation:
- z = (36,000 - 36,000) / 4,080 ≈ 0

Now that we have the z-scores for each life span, we can locate the corresponding percentiles on a standard normal distribution table or use a statistical software. The empirical rule states the following approximate percentiles based on the number of standard deviations away from the mean:

- Approximately 68% of the data falls within 1 standard deviation of the mean (z-scores between -1 and 1).
- Approximately 95% of the data falls within 2 standard deviations of the mean (z-scores between -2 and 2).
- Approximately 99.7% of the data falls within 3 standard deviations of the mean (z-scores between -3 and 3).

Using these guidelines, we can estimate the percentiles for each tire's life span:

1. Tire 1: z ≈ -1.13
- This z-score falls between -2 and -1, so the corresponding percentile is between 0.0228 (2.28%) and 0.1587 (15.87%). Let's use the midpoint, which is approximately 0.0907 (9.07%).

2. Tire 2: z ≈ 1.13
- This z-score falls between 1 and 2, so the corresponding percentile is between 0.8413 (84.13%) and 0.9772 (97.72%). Let's use the midpoint, which is approximately 0.9092 (90.92%).

3. Tire 3: z ≈ 0
- This z-score is approximately 0, so the corresponding percentile is 0.5 (50%).

Therefore, the estimated percentiles corresponding to the life spans of the three randomly selected tires are approximately:
1. Tire 1: 9.07%
2. Tire 2: 90.92%
3. Tire 3: 50%

To find the percentiles corresponding to the life spans using the empirical rule, we need to calculate the z-scores for each tire's distance traveled.

The z-score measures the number of standard deviations a data point is from the mean. We use the formula:

z = (x - μ) / σ

where:
- x is the individual data point
- μ is the mean of all data points
- σ is the standard deviation of all data points

Step 1: Calculate the mean (μ) of the tire life spans.
μ = (31,400 + 40,600 + 36,000) / 3
= 108,000 / 3
= 36,000 miles

Step 2: Calculate the standard deviation (σ) of the tire life spans.
To find the standard deviation, we need to calculate the variance first.
Variance = [(31,400 - 36,000)^2 + (40,600 - 36,000)^2 + (36,000 - 36,000)^2] / 3
= (15,600^2 + 4,600^2 + 0^2) / 3
= (243,360,000 + 21,160,000) / 3
= 264,520,000 / 3
≈ 88,173,333.3

Standard deviation (σ) = √Variance
= √88,173,333.3
≈ 9,384.33 miles

Step 3: Calculate the z-scores for each tire's distance traveled.
For the first tire:
z = (31,400 - 36,000) / 9,384.33
≈ -0.49

For the second tire:
z = (40,600 - 36,000) / 9,384.33
≈ 0.49

For the third tire:
z = (36,000 - 36,000) / 9,384.33
= 0

Step 4: Find the percentiles using the z-scores.
Using a standard Normal Distribution table, we can find the percentiles corresponding to the z-scores.

For the first tire:
The z-score is approximately -0.49, which corresponds to a percentile of approximately 31.89%.

For the second tire:
The z-score is approximately 0.49, which corresponds to a percentile of approximately 68.11%.

For the third tire:
The z-score is 0, which corresponds to a percentile of 50%.

Therefore, the percentiles that correspond to each tire's life span are approximately:
- First tire: 31.89%
- Second tire: 68.11%
- Third tire: 50%