What is the probability that the sample mean is between 45 and 52 minutes?
3) The amount of time required for an oi and filter change on an automobile is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample of 16 cars is selected.
Since the data is assumed to be normally distributed, the sampling mean equals the mean of the data. `mu_(bar(x))=mu` .
The standard deviation of the sampling distribution is the standard deviation of the data set divided by the square root of the sample size. Thus `sigma_(bar(x))=sigma/sqrt(n)`
Here `mu_(bar(x))=mu=45` and `sigma_(bar(x))=sigma/sqrt(n)=10/sqrt(16)=5/2` .
Now we can convert each score to a `z` score; converting (45,52) to `(z_1,z_2)` :
Then the probability of a score lying between `z_1=0"and"z_2=2.8` can be found by finding the area under the normal curve between these scores.
The area to the left of `z_1=0` is .5 and the area to the left of `z_2=2.8` is approximately .9974
So the area under the curve between these points is approximately .9974-.5=.4974
The probability of a random sample of 16 cars having a sample mean between 45 and 52 is 49.74%