The duration of time it takes General Motors to build a car is normally distributed with a mean of 31 hours and a standard deviation of 2 hours.
a) What is the probability that a single car, selected at random,
will take between 28 and 34 hours?
b) What is the probability that the mean of the times for a sample
of 25 cars will be between 28 and 34 hours?
a) Z = (score-mean)/SD
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.
b) Z = (score-mean)/SEm
SEm = SD/√n
Use the table.
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To answer these questions, we need to use the concept of the normal distribution and calculate the z-scores. The z-score measures how many standard deviations an observation is from the mean.
For part a), we want to know the probability that a single car will take between 28 and 34 hours.
1. Calculate the z-score for 28 hours:
z = (x - μ) / σ
where x is the value (28 hours), μ is the mean (31 hours), and σ is the standard deviation (2 hours).
z = (28 - 31) / 2 = -1.5
2. Calculate the z-score for 34 hours:
z = (x - μ) / σ
where x is the value (34 hours), μ is the mean (31 hours), and σ is the standard deviation (2 hours).
z = (34 - 31) / 2 = 1.5
3. Look up the z-score probabilities in the standard normal distribution table (also known as the z-table) or use a calculator. The probability of a z-score being between -1.5 and 1.5 is approximately 0.8664.
Therefore, the probability that a single car will take between 28 and 34 hours is approximately 0.8664.
For part b), we want to know the probability that the mean of the times for a sample of 25 cars will be between 28 and 34 hours.
1. Calculate the standard error of the mean (SE):
SE = σ / √n
where σ is the standard deviation (2 hours) and n is the sample size (25).
SE = 2 / √25 = 2 / 5 = 0.4
2. Calculate the z-score for 28 hours:
z = (x - μ) / SE
where x is the value (28 hours), μ is the mean (31 hours), and SE is the standard error of the mean (0.4).
z = (28 - 31) / 0.4 = -7.5
3. Calculate the z-score for 34 hours:
z = (x - μ) / SE
where x is the value (34 hours), μ is the mean (31 hours), and SE is the standard error of the mean (0.4).
z = (34 - 31) / 0.4 = 7.5
4. Again, look up the z-score probabilities in the standard normal distribution table or use a calculator. The probability of a z-score being between -7.5 and 7.5 is practically 1.
Therefore, the probability that the mean of the times for a sample of 25 cars will be between 28 and 34 hours is approximately 1.