It is estimated that amounts of money spent on gasoline by customers at their gas stations follows a normal distribution witha standard deviation of $3. It was also found that 10% of all customers spent more than $25. What percentage of customers spent less than $20.
Z = (score-mean)/SD = (25-mean)/3
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.1) that corresponds to a Z score.
Use that Z score in the above equation to find the mean. Then use that mean and Z score to find the new percentage.
To find the percentage of customers who spent less than $20, we need to calculate the z-score corresponding to $20, using the standard deviation and mean of the normal distribution.
The z-score formula is given by:
z = (x - μ) / σ
Where:
z = z-score
x = value we want to find the probability for ($20 in this case)
μ = mean of the distribution
σ = standard deviation of the distribution
Since we are not given the mean (μ) of the distribution, we cannot directly find the z-score. However, we are given a percentage (10%) along with the standard deviation.
To find the z-score corresponding to the given percentage (10%), we can use a standard normal distribution table or a statistical calculator.
Using either method, we find that the z-score corresponding to a cumulative percentage of 10% is approximately -1.28.
Now we can solve for the mean (μ) using the z-score formula:
-1.28 = (20 - μ) / 3
Rearranging the equation, we get:
-3.84 = 20 - μ
Solving for μ, we find:
μ ≈ 20 + 3.84 ≈ 23.84
Therefore, the mean (μ) of the distribution is approximately $23.84.
Now that we have both the mean (μ) and the standard deviation (σ), we can find the z-score for $20 using the z-score formula:
z = (20 - 23.84) / 3
Calculating this, we get:
z ≈ -1.28
Finally, we can use a standard normal distribution table or a statistical calculator to find the cumulative percentage (probability) corresponding to a z-score of -1.28.
The table or calculator will give us the percentage of customers who spent less than $20, which is approximately 10.00%.
Therefore, approximately 10% of customers spent less than $20.