An interval estimate for the average distance car tyres lasted during their "lifetime" was reported to be 34578 kilometres to 38512 kilometres. This interval estimate was based on a sample of 46 tyres. The variance of the lifetime was determined from previous studies to be 48512333 km squared. What level of confidence can be attributed to this interval estimate? State your answer as a percentage, correct to the nearest whole number.
Question
An interval estimate for the average distance car tyres lasted during their "lifetime" was reported to be 34578 kilometres to 38512 kilometres. This interval estimate was based on a sample of 46 tyres. The variance of the lifetime was determined from previous studies to be 48512333 km squared. What level of confidence can be attributed to this interval estimate? State your answer as a percentage, correct to the nearest whole number.
Solution
To solve this problem, we need to use the formula for a confidence interval for a population mean when the population standard deviation is known. The formula is:
X̄ ± Z * (σ/√n)
where:
- X̄ is the sample mean,
- Z is the Z-score (which corresponds to the desired level of confidence),
- σ is the population standard deviation, and
- n is the sample size.
First, we need to calculate the sample mean (X̄), which is the midpoint of the interval estimate. The interval estimate is from 34578 km to 38512 km, so the sample mean is:
X̄ = (34578 + 38512) / 2 = 36545 km
Next, we need to calculate the population standard deviation (σ). We are given the population variance, which is the square of the standard deviation. The variance is 48512333 km squared, so the standard deviation is the square root of this:
σ = √48512333 = 6965 km
The sample size (n) is given as 46.
The width of the confidence interval is the difference between the upper limit of the interval estimate and the sample mean, which is:
38512 - 36545 = 1967 km
We can now set up the equation for the confidence interval and solve for the Z-score:
1967 = Z * (6965/√46)
Solving for Z gives:
Z = 1967 / (6965/√46) = 1.96
The Z-score of 1.96 corresponds to a confidence level of approximately 95%. Therefore, we can be 95% confident that the average distance car tyres lasted during their "lifetime" is between 34578 kilometres and 38512 kilometres.
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