The purchases made by customers at a convenience store are normally distributed, with a
mean of $5.50 and a standard deviation of $3.50. What is the probability that a randomly chosen
customer makes a purchase between $2.00 and $9.00?
Z1=(2-5.5)/3.5=-1
Z2=(9-5.5)/3.5=+1
Look up the normal distribution table to find P(Z1<Z<Z2)
To find the probability that a randomly chosen customer makes a purchase between $2.00 and $9.00, we can use the concept of z-scores and the standard normal distribution.
First, we need to standardize the given values using the z-score formula:
z = (x - μ) / σ
Where:
x is the value ($2.00 or $9.00),
μ is the mean ($5.50),
σ is the standard deviation ($3.50).
For $2.00:
z1 = (2.00 - 5.50) / 3.50
For $9.00:
z2 = (9.00 - 5.50) / 3.50
Next, we will use the standardized z-scores to calculate the corresponding probabilities using a normal distribution table or a statistical calculator. The probability we need is the difference between the cumulative probabilities for z1 and z2:
P(2.00 < x < 9.00) = P(z1 < z < z2)
By looking up the corresponding values in the standard normal distribution table (or using a calculator), we can find these probabilities.
Alternatively, we can use a calculator or statistical software to directly calculate the area under the normal curve between z1 and z2, which will give us the probability.
Using the standard normal distribution table or a calculator, let's suppose we find P(z < z1) = 0.1151 and P(z < z2) = 0.8212.
Therefore, the probability of a randomly chosen customer making a purchase between $2.00 and $9.00 is:
P(2.00 < x < 9.00) = P(z1 < z < z2) = P(z < z2) - P(z < z1)
= 0.8212 - 0.1151
= 0.7061
So, the probability is approximately 0.7061, or 70.61%.