The life spans of three randomly selected tires are 31,950 miles, 36,050 miles, 34,000 miles. Using the empirical rule, find the percentile that corresponds to each life span.

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See Quala's post.

To find the percentile that corresponds to each life span using the empirical rule, we need to calculate the z-score for each value and then use a z-table to determine the corresponding percentile.

The empirical rule states that for a normally distributed data set:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.

Let's calculate the z-scores for each life span value:
1. Calculate the mean (average) of the data set:
Mean = (31,950 + 36,050 + 34,000) / 3 = 34,666.67 miles (rounded to nearest whole number)

2. Calculate the standard deviation of the data set:
Standard Deviation = square root of [(sum of (data value - mean)^2) / number of data points]
Standard Deviation = square root of [((31,950 - 34,666.67)^2 + (36,050 - 34,666.67)^2 + (34,000 - 34,666.67)^2) / 3]
Standard Deviation = square root of [(738,555,555.56 + 13,555,555.56 + 489,555.56) / 3]
Standard Deviation = square root of [752,600,555.56 / 3]
Standard Deviation = square root of 250,866,851.85
Standard Deviation ≈ 15,845.73 miles (rounded to nearest hundredth)

3. Now, calculate the z-score for each life span value using the formula:
z = (data value - mean) / standard deviation

For the first tire's life span (31,950 miles):
z = (31,950 - 34,666.67) / 15,845.73
z ≈ -0.171

For the second tire's life span (36,050 miles):
z = (36,050 - 34,666.67) / 15,845.73
z ≈ 0.087

For the third tire's life span (34,000 miles):
z = (34,000 - 34,666.67) / 15,845.73
z ≈ -0.042

4. Now, we need to use a z-table to determine the percentile corresponding to each z-score. A z-table provides an area (probability) associated with each z-score.

For the first tire's life span (z ≈ -0.171):
The z-table gives us the area/probability to the left of the z-score.
From the table, we find that the area/probability to the left of -0.171 is approximately 0.4325.
Therefore, the first tire's life span corresponds to the 43.25th percentile.

For the second tire's life span (z ≈ 0.087):
Using the z-table, we find that the area/probability to the left of 0.087 is approximately 0.5331.
Therefore, the second tire's life span corresponds to the 53.31st percentile.

For the third tire's life span (z ≈ -0.042):
Using the z-table, we find that the area/probability to the left of -0.042 is approximately 0.4821.
Therefore, the third tire's life span corresponds to the 48.21st percentile.

So, the percentile that corresponds to each life span using the empirical rule are:
- First tire's life span: 43.25th percentile
- Second tire's life span: 53.31st percentile
- Third tire's life span: 48.21st percentile.