Answer:
a. The interval: 26,978 miles and 32,022 miles
b. The interval: 26,000 miles and 32,000 miles
Explanation:
a. It can be guaranteed that 75% of the lifetimes of tires of this brand will be in what interval?
0.674
We solve question a, using z score formula.
z = (x-μ)/σ, where
x is the raw score
μ is the population mean = 29,000 miles
σ is the population standard deviation = 3,000 miles
We are asked to find the interval = x
z = z score of 75th and 25th percentile = ±0.674
Hence:
For z = -0.674
-0.674 = x - 29,000/3,000
Cross Multiply
-0.674 × 3,000 = x - 29,000
-2022 = x - 29,000
x = - 2022 + 29,000
x = 26,978 miles
For z = 0.674
0.674 = x - 29,000/3,000
Cross Multiply
0.674 × 3,000 = x - 29,000
2022 = x - 29,000
x = 2022 + 29,000
x = 31,022 miles
The interval: 26,978 miles and 32,022 miles
b. Using the empirical rule, it can be estimated that approximately 68% of the lifetimes of tires of this brand will be in what interval?
68% of data falls within 1 standard deviations from the mean - between μ – σ and μ + σ .
Mean (μ) = 29,000 miles
Standard deviation (σ) = 3,000 miles.
Hence the interval is calculated as:
μ – σ
= 29,000 - 3,000
= 26,000 miles
μ + σ
= 29,000 + 3,000
= 32,000 miles
The interval: 26,000 miles and 32,000 miles