Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z scores.
Between 36 and 40 passengers
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z scores.
Step 1: Convert the individual values to z-scores using the formula:
z = (x - μ) / σ
Where:
x = individual value
μ = mean
σ = standard deviation
For 36 passengers:
z1 = (36 - 48) / 3 = -4. Therefore, z1 = -4.
For 40 passengers:
z2 = (40 - 48) / 3 = -2.67. Therefore, z2 ≈ -2.67.
Step 2: Use the z-scores to find the corresponding probabilities using a standard normal distribution table or calculator.
P(36 < X < 40) = P(-4 < Z < -2.67)
Approximately, P(-4 < Z < -2.67) = P(Z < -2.67) - P(Z < -4)
Look up the z-scores in the standard normal distribution table to find the corresponding probabilities.
P(Z < -2.67) ≈ 0.0034
P(Z < -4) ≈ 0.00003
Step 3: Calculate the final probability by subtracting P(Z < -4) from P(Z < -2.67).
P(-4 < Z < -2.67) ≈ 0.0034 - 0.00003
So, the probability that the express bus will have between 36 and 40 passengers is approximately 0.00337.
Step 1: Convert the given values to z-scores.
To find the z-score for each value, we can use the formula: (x - μ) / σ
where x is the given value, μ is the mean, and σ is the standard deviation.
For 36 passengers:
z-score = (36 - 48) / 3 = -4
For 40 passengers:
z-score = (40 - 48) / 3 = -2.67
Step 2: Find the probability associated with each z-score using the standard normal distribution table.
The standard normal distribution table provides the area under the curve to the left of a given z-score.
Using the table, we find that the probability associated with a z-score of -4 is approximately 0.00003168.
Using the table, we find that the probability associated with a z-score of -2.67 is approximately 0.00383.
Step 3: Calculate the probability between the two z-scores.
To find the probability between two z-scores, we subtract the probability associated with the lower z-score from the probability associated with the higher z-score.
Probability between 36 and 40 passengers = Probability (z-score ≤ -2.67) - Probability (z-score ≤ -4)
= 0.00383 - 0.00003168
= 0.00379832
Therefore, the probability that the bus will have between 36 and 40 passengers is approximately 0.0038, or 0.38%.