A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Cavalier shows that the mean is 4.5 hours and the standard deviation is .6 hours. If 42 mechanics are randomly selected, find the probability that their mean rebuild time is less than 3.5 hours.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
I'm not quite sure I understand. What do I use as the score, and what do I use for n.
To find the probability that the mean rebuild time is less than 3.5 hours, we can use the central limit theorem.
The central limit theorem states that for a large enough sample size (in this case, 42 mechanics) from a population with any distribution, the distribution of the sample means will approximate a normal distribution.
To apply the central limit theorem, we need to find the mean and standard deviation of the sampling distribution of the mean.
The mean of the sampling distribution of the mean (also known as the population mean) is equal to the mean of the population, which is given as 4.5 hours.
The standard deviation of the sampling distribution of the mean (also known as the standard error) is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard deviation of the population is 0.6 hours, and the sample size is 42.
Standard error = standard deviation of the population / square root of sample size
= 0.6 / √42
≈ 0.0828 hours
Now that we have the mean and standard deviation of the sampling distribution, we can use the Z-score formula to find the probability.
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value we want to find the probability for (in this case, 3.5 hours)
μ is the population mean (4.5 hours)
σ is the standard deviation of the sampling distribution (0.0828 hours)
Z = (3.5 - 4.5) / 0.0828
= -12.08
Next, we need to find the probability associated with this Z-score using a standard normal distribution table or a calculator.
Looking up the Z-score of -12.08 in a standard normal distribution table, we find that the probability is approximately 0.0.
Therefore, the probability that the mean rebuild time is less than 3.5 hours for a sample of 42 mechanics is approximately 0.0.