Distance travelled in a week by sales staff employed by nationwide security firm is approximately normally distributed with a mean of miles and standard deviation of 48 miles. Estimate the percentage of staff who in a week travel less than 80 miles
To estimate the percentage of staff who travel less than 80 miles in a week, we can use the normal distribution and the given information about the mean and standard deviation.
Step 1: Standardize the value
We need to find the standardized value (z-score) for 80 miles, which represents how many standard deviations away from the mean this value is. The z-score can be calculated using the formula:
z = (x - μ) / σ
Where:
x is the value (80 miles),
μ is the mean (given value),
σ is the standard deviation (given value).
In this case,
x = 80 miles,
μ = mean (given value),
σ = standard deviation (given value).
Step 2: Find the area under the curve
Once we have the z-score, we can find the area under the standard normal curve corresponding to that value. This area represents the percentage of staff who travel less than 80 miles.
Step 3: Look up the area in the z-table
We can use a z-table or a statistical calculator to find the area under the curve. The z-table gives us the area to the left of the z-score. Simply find the z-score in the table or use a statistical calculator to determine the area.
Step 4: Calculate the percentage
The area under the curve corresponds to a percentage. Multiply this percentage by 100 to get the estimated percentage of staff who travel less than 80 miles in a week.
Using the given mean and standard deviation, the steps above will help estimate the percentage of staff who travel less than 80 miles in a week.