SOLUTION: A sample of 100 tyres is taken from a lot. The mean life of tyres is found to be 39350 kilometres with a standard deviation of 3260 kilometres. Could the sample come from a populat

Algebra ->  Probability-and-statistics -> SOLUTION: A sample of 100 tyres is taken from a lot. The mean life of tyres is found to be 39350 kilometres with a standard deviation of 3260 kilometres. Could the sample come from a populat      Log On


   

Question 1142169: A sample of 100 tyres is taken from a lot. The mean life of tyres is found to be 39350 kilometres with a standard deviation of 3260 kilometres. Could the sample come from a population with mean life of 40000 kilometres? Also obtain 99% confidence limits within which the mean life of tyres is expected to lie.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
sample size is 100 tires.
mean of sample is 39350
standard deviation of sample is 3260

standard error of the distribution of sample means is 3260 / sqrt(100) = 326.

because your standard deviation is taken from the sample rather than from the population, you would use the t-score rather than the z-score.

because the sample size is large enough, it shouldn't make much difference whether you use the t-score or the z-score.

at 99% confidence limits, the critical z-score would be plus or minus 2.575829303 and the critical t-score with 99 degrees of freedom would be plus or minus 2.62640545.

the basic formula for z-score or t-score is z or t = (x - m) / s

z = s-score
t = t-score
x is the raw score being compared to the mean.
m is the mean
s is the standard error of the distribution of sample means.

x is the mean of the sample = 39350
m is the mean of the population = 40000
s is the standard error of the distribution of sample means = 326.

the z-score / t-score would be equal to (39350 - 40000) / 326 = -1.993865031.

this score is well within the confidence limits of either the z-score or the t-score.

therefore, it's safe to say that this sample could come from a population that had a mean tire life of 40,000.

visually, the results look like this for the z-score.

$$$

note that the area to the left of a raw score of 39350 is equal to .023.....

since the area to the left of the critical z-score would be .005, 39350 is within the confidence limits.