The amount of time it takes to take your order at the local Whaters restaurant has a population mean m of 3.1 minutes and a standard deviation s of 0.40 minutes. If you select a random sample of 16 customers:
Q-3a: What is the probability that the mean time spent per customer is at least 3 minutes?
It is not mentioned that the distribution is normal. We will assume that it is, either by assumption, or by an approximation using the central limit theorem (which preferably applies to samples of 30+).
With the assumption, we can see that the variance is σ²/n=0.4²/16=0.01, hence σ=0.1.
Look up the Normal distribution table for 1-tail probability of 1σ which gives the probability of 1-0.8413=0.1587 for less than μ-1σ=3.1-0.1=3.0 minutes.
Can you take it from here?
No
To find the probability that the mean time spent per customer is at least 3 minutes, we can use the central limit theorem and the standard normal distribution.
1. Calculate the standard error of the mean (SE):
The formula for the standard error of the mean is SE = s / sqrt(n), where:
- s is the standard deviation of the population (0.40 minutes in this case), and
- n is the sample size (16 customers in this case).
Therefore, SE = 0.40 / sqrt(16) = 0.10 minutes.
2. Standardize the value of interest:
To standardize the value (3 minutes), we subtract the population mean (m) from the value of interest and then divide by the standard error (SE):
Z = (3 - 3.1) / 0.10 = -0.1 / 0.10 = -1.
3. Find the probability using the standard normal distribution table:
We want to find the probability that the mean time spent per customer is at least 3 minutes. Since our Z-value is negative (-1), we want to find the area to the left of this value in the standard normal distribution table.
By looking up the Z-value of -1 in the standard normal distribution table, we find that the corresponding area to the left of -1 is approximately 0.1587.
4. Calculate the probability of at least 3 minutes:
Since we're interested in the probability of at least 3 minutes, we subtract the area to the left of -1 from 1:
Probability = 1 - 0.1587 = 0.8413.
Therefore, the probability that the mean time spent per customer is at least 3 minutes is approximately 0.8413, or 84.13%.