a study of the amount of time it takes a mechanic to rebuild the transmission of a 1992 chevrolet cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. if 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 8.7 hours.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
Since only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to that Z score.
To find the probability that the mean rebuild time exceeds 8.7 hours for a sample of 40 mechanics, we can use the Central Limit Theorem. The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
In this case, we know that the population mean (μ) is 8.4 hours and the population standard deviation (σ) is 1.8 hours. Since the sample size is large (n = 40), we can assume that the sample mean follows a normal distribution.
To calculate the probability, we need to standardize the value 8.7 using the formula for z-score:
z = (x - μ) / (σ / sqrt(n))
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Plugging in the values:
z = (8.7 - 8.4) / (1.8 / sqrt(40))
z = 0.3 / (1.8 / 6.324)
Now, we can calculate the probability using a standard normal distribution table or a calculator.
Using a standard normal distribution table, the probability can be found by looking up the corresponding z-score. In this case, we want to find the probability that the z-score is greater than 0.3.
By looking up the z-score in the table, we find that the probability is approximately 0.382 (or 38.2%).
Therefore, the probability that the mean rebuild time exceeds 8.7 hours for a sample of 40 mechanics is approximately 0.382 (or 38.2%).