A luxury passenger liner has 500 passengers on board whose ages are normally distributed around a mean of 60 years with a standard deviation of 12 years. How many passengers are between 45 and 78 years old?
To solve this, we need to find the percentage of passengers whose ages are between 45 and 78 years old.
First, we'll standardize the lower and upper limits of the age range using the z-score formula:
For the lower limit of 45 years old:
z1 = (45 - 60) / 12 = -1.25
For the upper limit of 78 years old:
z2 = (78 - 60) / 12 = 1.5
Next, we'll find the corresponding percentage for each z-score using a standard normal distribution table or a calculator.
For the lower limit:
P(Z < -1.25) ≈ 0.1056 (approximately)
For the upper limit:
P(Z < 1.5) ≈ 0.9332 (approximately)
To find the percentage of passengers between the two limits, we subtract the lower limit percentage from the upper limit percentage:
P(-1.25 < Z < 1.5) = 0.9332 - 0.1056 = 0.8276
Finally, we multiply the percentage by the total number of passengers:
0.8276 * 500 ≈ 413.8
Approximately 414 passengers are between 45 and 78 years old.
To find the number of passengers between 45 and 78 years old, we need to calculate the area under the Normal distribution curve between these two ages.
Step 1: Calculate the Z-scores for the ages of 45 and 78. The Z-score formula is given by:
Z = (X - μ) / σ
where:
X = individual age
μ = mean age
σ = standard deviation of ages
Z1 = (45 - 60) / 12
Z2 = (78 - 60) / 12
Step 2: Look up the corresponding areas under the Normal distribution curve for the calculated Z-scores using a Z-table or a calculator.
Step 3: Calculate the area between these two Z-scores by subtracting the smaller area from the larger one.
Step 4: Finally, multiply the calculated area by the total number of passengers (500) to find the number of passengers between 45 and 78 years old.
Let's calculate these steps:
Step 1:
Z1 = (45 - 60) / 12 = -1.25
Z2 = (78 - 60) / 12 = 1.5
Step 2:
Using a Z-table or calculator, the area corresponding to Z1 = -1.25 is approximately 0.1056, and the area corresponding to Z2 = 1.5 is approximately 0.9332.
Step 3:
Area between Z1 and Z2 = 0.9332 - 0.1056 = 0.8276
Step 4:
Number of passengers between 45 and 78 years old = 0.8276 * 500 = 413.8
Therefore, approximately 413 passengers are between 45 and 78 years old on the luxury passenger liner.
To find the number of passengers between 45 and 78 years old, we need to calculate the proportion of passengers within this age range.
First, we need to calculate the z-scores for the given ages using the mean and standard deviation. The z-score formula is:
z = (x - μ) / σ
where:
- z is the z-score,
- x is the given age,
- μ is the mean age, and
- σ is the standard deviation.
For 45 years old:
z = (45 - 60) / 12
z = -1.25
For 78 years old:
z = (78 - 60) / 12
z = 1.5
Next, we can refer to the z-score table or use a calculator to find the corresponding probabilities associated with these z-scores. The table or calculator will give us the proportion of the normal distribution that falls between these two z-scores.
Using either method, we find that the proportion of passengers between -1.25 and 1.5 is approximately 0.7665.
Finally, to find the number of passengers within this age range, we multiply the proportion by the total number of passengers on board:
Number of passengers = proportion * total number of passengers
Number of passengers = 0.7665 * 500
Number of passengers ≈ 383
Therefore, approximately 383 passengers on the luxury liner are between 45 and 78 years old.