Drivers pay an average of (mean $690) per year for automobile insurance the distribution of insurance payments is approximately normal with a standard deviation of 110 dollars. What proportion of drivers pay more than 100 dollars per year for insurance?
Use z-scores.
z = (x - mean)/sd
x = 100
mean = 690
sd = 110
Plug the values into the formula, then use a z-table to determine the proportion.
I hope this will help get you started.
To find the proportion of drivers who pay more than $100 per year for insurance, we will use the normal distribution.
Step 1: Standardize the value
To use the normal distribution, we need to standardize the value of $100. Standardizing means transforming the value using the formula:
Z = (X - μ) / σ
where Z is the standardized value, X is the value we want to standardize, μ is the mean of the distribution, and σ is the standard deviation of the distribution.
In this case, X = $100, μ = $690, and σ = $110. Plugging these values into the formula, we get:
Z = (100 - 690) / 110
Step 2: Calculate the Z-score
Calculating this, we find:
Z = -5.36
Step 3: Find the proportion
Now we need to find the proportion of drivers who pay more than $100 per year for insurance. This is equivalent to finding the area to the right of the standardized value on the standard normal distribution table.
Looking up the Z-score of -5.36 on a standard normal distribution table (or using a calculator or statistical software), we find that the area to the right of -5.36 is very close to 1.
Therefore, approximately 100% of drivers pay more than $100 per year for insurance.
Alternatively, if you are using a calculator or statistical software, you can use the cumulative distribution function (CDF) of the normal distribution to find the proportion directly. The CDF will give you the area to the left of a specific value, so you would use the complementary probability (1 - CDF) to find the area to the right of the value.