The number of customers who enter a supermarket each hour is normally distributed with a mean of 600 and a standard deviation of 200. The supermarket is open 16 hours per day. what is the probability that the total number of customers who enter the supermarket in one day is greater than 10,000. (Hint,Calculate the average hourly number of customers necessary to exceed 10,000 in one 16-hour day)
To calculate the probability that the total number of customers who enter the supermarket in one day is greater than 10,000, we first need to calculate the average hourly number of customers necessary to exceed 10,000 in a 16-hour day.
Since the mean number of customers per hour is 600, we can calculate the average hourly number of customers as follows:
Average Hourly Number of Customers = Total Number of Customers / Total Hours
Total Number of Customers = 10,000
Total Hours = 16
Average Hourly Number of Customers = 10,000 / 16 = 625
Now that we know the average hourly number of customers needed to exceed 10,000 in a day, we can calculate the probability using the normal distribution.
We need to calculate the z-score for the average hourly number of customers. Using the formula:
z = (x - μ) / σ
where:
x = average hourly number of customers
μ = mean number of customers per hour (600)
σ = standard deviation of number of customers per hour (200)
z = (625 - 600) / 200 = 0.125
Next, we need to find the probability of getting a value greater than the z-score we just calculated using a standard normal distribution table or a statistical calculator.
Using a standard normal distribution table or a calculator, we find that the probability of getting a value greater than 0.125 is approximately 0.4505.
Therefore, the probability that the total number of customers who enter the supermarket in one day is greater than 10,000 is approximately 0.4505 or 45.05%.
To solve this problem, we need to calculate the average hourly number of customers necessary to exceed 10,000 in one 16-hour day. Then, we can use the normal distribution to find the probability.
Step 1: Calculate the average number of customers per hour required to reach 10,000 customers in 16 hours.
The total number of customers needed in 16 hours is 10,000. Therefore, the average number of customers per hour is calculated by dividing 10,000 by 16:
Average hourly number of customers = 10,000 / 16 = 625
So, in order to have a total of more than 10,000 customers in one day, the average number of customers per hour must be greater than 625.
Step 2: Calculate the z-score.
The z-score represents the number of standard deviations a value is from the mean. In this case, we need to calculate the z-score for a value greater than 625, using the mean and standard deviation provided.
z = (x - mean) / standard deviation
Here, x is the value we want to calculate the probability for (625), mean is the mean number of customers per hour (600), and the standard deviation is the standard deviation of the number of customers per hour (200).
z = (625 - 600) / 200 = 0.125
Step 3: Find the probability using a z-table or calculator.
Using a z-table or calculator, we can find the probability corresponding to the calculated z-score of 0.125. The probability of having a value greater than 625 can be determined by subtracting the cumulative probability from 0.5 (since the normal distribution is symmetric).
P(x > 625) = 1 - P(z ≤ 0.125)
Using a z-table or calculator, we can find the cumulative probability of 0.125, and subtract it from 1 to get the probability.
Step 4: Calculate the final probability.
Once you have obtained the cumulative probability from the z-table or calculator, subtract it from 1 to obtain the final probability.
P(x > 625) = 1 - cumulative probability obtained from the z-table or calculator
This will give you the probability that the total number of customers who enter the supermarket in one day is greater than 10,000.
This is an odd supermarket, having even traffic all day. Most would kill for this even traffic flow.
number per hour= 10,000/16=625
standard deviation 200
number standard deviations: 25/200
Look that up in your tables, or here: