A large metropolitan area found that the time it takes for people to commute to work has a mean of 20.5 minutes and a standard deviation of 15.4 minutes. What is the probability that a random sample of 40 people have a mean commute time greater than 25 minutes?
z-score = (25-20.5)/15.4
then look up that z in your tables,
subtract that from 1
that is,
1 - value you looked up
or
you can use one of my favourite websites
http://davidmlane.com/normal.html
Make sure "area from value " is clicked on
enter the mean, the sd, and click on "below" and enter 25
The value of .6149 should agree with the value you found above
To find the probability that a random sample of 40 people have a mean commute time greater than 25 minutes, we can use the Central Limit Theorem.
The Central Limit Theorem states that the distribution of sample means from a population with any distribution approaches a normal distribution as the sample size increases.
First, let's calculate the standard error of the sample mean using the formula:
Standard Error = Standard Deviation / √(Sample Size)
Standard Error = 15.4 / √(40)
Standard Error ≈ 2.437
Next, we need to standardize the sample mean using the Z-score formula:
Z = (X - Mean) / Standard Error
Z = (25 - 20.5) / 2.437
Z ≈ 1.84
Now, we need to find the probability that the Z-score is greater than 1.84. We can do this by looking up the value in the standard normal distribution table or using a statistical calculator.
From the standard normal distribution table, a Z-score of 1.84 corresponds to a probability of approximately 0.9664.
Therefore, the probability that a random sample of 40 people have a mean commute time greater than 25 minutes is approximately 0.9664 or 96.64%.