Distance travelled in a week by sales staff employed by nationwide security firm is approximately normally distributed with a mean of 200 miles and standard deviation of 48 miles. Estimate the percentage of staff who in a week travel:
1.
a)less than 80 miles
b)more than 250 miles
c)between 100 and 300 miles
2.A reduced mileage rate is to be introduced which is to be set such that approximately 20% of the sales staff will be affected? What is the maximum miles per week that a member of staff could travel before reaching the reduced mileage rate?
3)
Complete this sentence:
"80% of sales staff travel between ____ and ______ miles each week".
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To estimate the percentage of staff that travel certain distances in a week, we will use the given mean and standard deviation of the distance travelled. From there, we will calculate the z-scores and use the z-scores to estimate the desired probabilities.
1a) To estimate the percentage of staff who travel less than 80 miles in a week, we need to find the cumulative probability to the left of 80.
First, we calculate the z-score for 80 miles using the formula:
z = (x - mean) / standard deviation
z = (80 - 200) / 48
z ≈ -2.083
We can then use a standard normal distribution table or a calculator to find the cumulative probability corresponding to a z-score of -2.083. The cumulative probability is approximately 0.0188 or 1.88%.
Therefore, approximately 1.88% of the staff travel less than 80 miles in a week.
1b) To estimate the percentage of staff who travel more than 250 miles in a week, we need to find the cumulative probability to the right of 250.
First, we calculate the z-score for 250 miles:
z = (x - mean) / standard deviation
z = (250 - 200) / 48
z ≈ 1.042
Using the standard normal distribution table or a calculator, we find the cumulative probability to the left of 1.042 and subtract it from 1 to get the cumulative probability to the right. The cumulative probability is approximately 0.1492 or 14.92%.
Therefore, approximately 14.92% of the staff travel more than 250 miles in a week.
1c) To estimate the percentage of staff who travel between 100 and 300 miles in a week, we need to find the cumulative probability between these two values.
First, we calculate the z-scores for 100 miles and 300 miles:
z1 = (100 - 200) / 48
z1 ≈ -2.083
z2 = (300 - 200) / 48
z2 ≈ 2.083
Using the standard normal distribution table or a calculator, we find the cumulative probabilities corresponding to z1 and z2. Then, we subtract the cumulative probability associated with z1 from the cumulative probability associated with z2 to get the desired range.
Let's call the cumulative probability for z1 P1 and the cumulative probability for z2 P2:
Cumulative probability between 100 and 300 miles = P2 - P1.
Therefore, the percentage of staff who travel between 100 and 300 miles in a week can be estimated using P2 - P1.
2) To find the maximum miles per week that a member of staff could travel before reaching the reduced mileage rate, we need to find the value corresponding to the 20th percentile of the distribution.
We can calculate the z-score corresponding to the 20th percentile using the cumulative probability associated with the 20th percentile, which is 0.20.
Let's call the z-score Z:
Z = invNorm(0.20)
Using a standard normal distribution table or a calculator, we find the z-score corresponding to the 20th percentile. We can then use the formula for z-score to find the maximum miles per week that a member of staff could travel before reaching the reduced mileage rate:
Maximum miles = (Z * standard deviation) + mean.
Therefore, we can estimate the maximum miles per week a staff member could travel before reaching the reduced mileage rate.
3) To complete the sentence "80% of sales staff travel between ____ and ______ miles each week", we need to find the range of miles that corresponds to the 80th percentile.
We can calculate the z-score corresponding to the 80th percentile using the cumulative probability associated with the 80th percentile, which is 0.80.
Let's call the z-score Z:
Z = invNorm(0.80)
Using a standard normal distribution table or a calculator, we find the z-score corresponding to the 80th percentile. We can then use the formula for z-score to find the range of miles:
Range = (Z * standard deviation) + mean
Therefore, we can estimate the range of miles that corresponds to the 80th percentile of the sales staff.