the courier service company has found that their time of parcels to client is normally distributed with a mean of 45 minutes and a standard deviation of 8 minutes. the probability that a randomly selected parcel will take less than 48 minutes to deliver is
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Play around with a bit, and get comfortable with this kind of question.
To find the probability that a randomly selected parcel will take less than 48 minutes to deliver, we need to calculate the z-score and then find the corresponding probability from the standard normal distribution table.
Step 1: Calculate the z-score.
The z-score formula is given by:
z = (X - μ) / σ
Where:
X is the value we want to find the probability for (48 minutes in this case)
μ is the mean (45 minutes)
σ is the standard deviation (8 minutes)
Substituting the values into the formula:
z = (48 - 45) / 8 = 0.375
Step 2: Find the probability using the z-score.
Using the standard normal distribution table, we can find the probability corresponding to the z-score of 0.375.
Looking at the table, the closest value we can find is 0.6499 (corresponding to z = 0.37), and the next closest is 0.6554 (corresponding to z = 0.38).
To estimate the probability for z = 0.375, we can estimate it as the average of the probabilities for z = 0.37 and z = 0.38.
Estimated probability = (0.6499 + 0.6554) / 2 = 0.65265 (approximately)
So, the probability that a randomly selected parcel will take less than 48 minutes to deliver is approximately 0.65265 or 65.27% (to two decimal places).
To find the probability that a randomly selected parcel will take less than 48 minutes to deliver, we need to use the standard normal distribution table (also known as the Z-table) or a calculator that can calculate normal distribution probabilities.
Step 1: Standardize the value using the standard normal distribution formula:
Z = (X - μ) / σ
Where:
Z: Standardized value
X: Given value
μ: Mean
σ: Standard deviation
In this case:
X = 48 minutes
μ = 45 minutes
σ = 8 minutes
So, we have:
Z = (48 - 45) / 8 = 3/8 = 0.375
Step 2: Use the Z-table or a calculator to find the probability. Looking up the value of Z = 0.375 in the Z-table, we find that the corresponding probability is 0.6480.
Therefore, the probability that a randomly selected parcel will take less than 48 minutes to deliver is 0.6480, or approximately 64.80%.