A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
(a) Find on how many days of the year (365 days) the daily sales can be expected to exceed 3900 litres.
To find how many days of the year the daily sales can be expected to exceed 3900 litres, we need to determine the probability of the daily sales exceeding this value.
Let's denote X as the daily sales in litres. Given that X is normally distributed with a mean of 4520 and a standard deviation of 560, we can use the z-score formula to standardize the value of 3900.
The z-score formula is:
z = (X - μ) / σ
where X is the value of interest, μ is the mean, and σ is the standard deviation.
Let's calculate the z-score for a daily sales value of 3900:
z = (3900 - 4520) / 560
z = -0.1107 (rounded to 4 decimal places)
Next, we need to find the area under the normal distribution curve to the right of this z-score. This represents the probability of the daily sales exceeding 3900 litres.
We can use a z-table or a statistical calculator to find this area. For simplicity, let's use an online calculator.
Using an online calculator, the area to the right of the z-score -0.1107 is approximately 0.5438 (rounded to 4 decimal places).
Therefore, the probability of the daily sales exceeding 3900 litres is approximately 0.5438.
To find the number of days out of 365 days where the daily sales can be expected to exceed 3900 litres, we multiply the probability by the total number of days:
Number of days = probability * total number of days
= 0.5438 * 365
≈ 198.71
Therefore, on approximately 199 days of the year (365 days), the daily sales can be expected to exceed 3900 litres.
To find the number of days on which the daily sales can be expected to exceed 3900 liters, we need to calculate the probability of the daily sales being greater than 3900 liters.
First, we need to convert the problem into a standard normal distribution. We can do this by using the formula for standardizing a variable:
Z = (X - μ) / σ
Where:
Z is the standard score or Z-score
X is the value we want to convert
μ is the mean of the distribution
σ is the standard deviation of the distribution
In this case, X = 3900, μ = 4520, and σ = 560.
Calculating the Z-score:
Z = (3900 - 4520) / 560
Z = -0.1071
Next, we need to find the area under the standard normal curve to the right of this Z-score. We can use a Z-table or a statistical calculator to find this value. For example, using a Z-table, we can find that the area to the left of the Z-score -0.1071 is 0.4564. Therefore, the area to the right is 1 - 0.4564 = 0.5436.
This means that the probability of the daily sales exceeding 3900 liters is 0.5436.
To find the number of days on which the daily sales can be expected to exceed 3900 liters over 365 days, we multiply the probability by the total number of days:
Number of days = Probabil
well, 3900 is 1.107σ below the mean.
Now just head to your Z table to interpret that as a probability, then multiply that by 365