on the daily run of an express bus, the average number of passengers is 48. the standard deviation is 3. Assume the variable is normally distributed. find the probability that the bus will have.

a) between 36 and 40 passengers
b) fewer than 42 passengers
c) More than 48 passengers
d) between 43 and 47 passengers

Step by step solution

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To find the probability for each scenario, we will use the concept of the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

To find the probability for a distribution with a given mean and standard deviation, we need to standardize the values using the z-score formula:

z = (x - mean) / standard deviation

Where:
z is the z-score
x is the value we want to find the probability for
mean is the mean of the distribution
standard deviation is the standard deviation of the distribution

We can then use the standardized z-score to look up the probabilities in a standard normal distribution table or use a statistical calculator.

a) Between 36 and 40 passengers:

First, we need to standardize the values:
z1 = (36 - 48) / 3 = -4
z2 = (40 - 48) / 3 = -2.67

Next, we use the standard normal distribution table or a statistical calculator to find the probabilities corresponding to these z-scores.

The probability for z-score -4 is very close to 0 (essentially 0).
The probability for z-score -2.67 is 0.0038.

To find the probability between 36 and 40 passengers, we subtract the probability for z = -2.67 from the probability for z = -4:

P(36 ≤ x ≤ 40) = P(z ≤ -2.67) - P(z ≤ -4)

b) Fewer than 42 passengers:

First, we need to standardize the value:
z = (42 - 48) / 3 = -2

We find the probability for z = -2, which is 0.0228.

To find the probability of fewer than 42 passengers, we use P(z ≤ -2).

c) More than 48 passengers:

To find the probability of more than 48 passengers, we need to calculate the probability for z > 0.
Since the standard normal distribution is symmetric, we know that P(z ≤ 0) = 0.5.
Therefore, P(z > 0) = 1 - P(z ≤ 0) = 1 - 0.5 = 0.5.

d) Between 43 and 47 passengers:

First, we need to standardize the values:
z1 = (43 - 48) / 3 = -1.67
z2 = (47 - 48) / 3 = -0.33

We find the probability for z1 and z2 using the standard normal distribution table or a statistical calculator.

To find the probability between 43 and 47 passengers, we subtract the probability for z = -0.33 from the probability for z = -1.67:

P(43 ≤ x ≤ 47) = P(z ≤ -0.33) - P(z ≤ -1.67)

Remember to use a standard normal distribution table or a statistical calculator to find the probabilities for the corresponding z-scores.

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