a) Approximately 82.20% of the tires will last between 30,750 miles and 38,300 miles.
b) Approximately 5.48% of the tires will last at least 39,000 miles.
c) Approximately 8.46% of the tires will fail to live up to the guarantee of lasting at least 30,750 miles.
d) Approximately 13,700 tires out of 250,000 produced will last at least 39,000 miles.
a) To determine the percent of tires that will last between 30,750 miles and 38,300 miles, we need to calculate the z-scores for these values and find the corresponding areas under the standard normal distribution curve.
First, let's calculate the z-score for 30,750 miles:
z = (x - μ) / σ
z = (30,750 - 35,000) / 2,500
z ≈ -1.38
Next, let's calculate the z-score for 38,300 miles:
z = (x - μ) / σ
z = (38,300 - 35,000) / 2,500
z ≈ 1.32
Using a standard normal distribution table or a calculator, we can find the areas corresponding to these z-scores. The area between these two z-scores represents the percentage of tires that will last between 30,750 miles and 38,300 miles.
The area to the left of z = -1.38 is approximately 0.0846.
The area to the left of z = 1.32 is approximately 0.9066.
To find the area between these two z-scores, we subtract the smaller area from the larger area:
0.9066 - 0.0846 ≈ 0.8220
So, approximately 82.20% of the tires will last between 30,750 miles and 38,300 miles.
b) To determine the percent of tires that will last at least 39,000 miles, we need to calculate the z-score for this value and find the area to the right of the z-score.
The z-score for 39,000 miles is:
z = (x - μ) / σ
z = (39,000 - 35,000) / 2,500
z = 1.60
The area to the right of z = 1.60 is approximately 0.0548.
So, approximately 5.48% of the tires will last at least 39,000 miles.
c) If the manufacturer guarantees the tires to last at least 30,750 miles, we need to find the percentage of tires that will fail to live up to the guarantee. This is the area to the left of 30,750 miles.
Using the z-score formula, we can calculate the z-score for 30,750 miles:
z = (x - μ) / σ
z = (30,750 - 35,000) / 2,500
z ≈ -1.38
The area to the left of z = -1.38 is approximately 0.0846.
So, approximately 8.46% of the tires will fail to live up to the guarantee.
d) To determine the number of tires that will last at least 39,000 miles out of 250,000 tires produced, we can multiply the percentage found in part b) by the total number of tires:
Number of tires = Percentage of tires * Total number of tires
= 0.0548 * 250,000
≈ 13,700 tires
Approximately 13,700 tires will last at least 39,000 miles.