If the height of 200 students is normally distributed with mean of 65 inches & std deviation 3.5 inches. How many students have height less than 5?
You can play around with Z table stuff at
davidmlane.com/hyperstat/z_table.html
To find the number of students with a height less than 5 inches, we need to calculate the z-score and then use a standard normal distribution table. The z-score measures how many standard deviations a particular value is from the mean.
First, we need to calculate the z-score for a height of 5 inches using the following formula:
z = (x - μ) / σ
Where:
- x is the height value (5 inches),
- μ is the mean (65 inches), and
- σ is the standard deviation (3.5 inches).
Plugging in the values, we get:
z = (5 - 65) / 3.5
z = -60 / 3.5
z ≈ -17.14
Now, we can use a standard normal distribution table to find the proportion of students with a z-score less than -17.14. Looking up the z-score in the table, we find that the area/proportion to the left of z = -17.14 is virtually zero. This means that almost none of the students have a height less than 5 inches.
So, the answer is practically zero students have a height less than 5 inches.
To find out how many students have a height less than 5 inches, we need to calculate the z-score for a height of 5 inches and then use a standard normal distribution table to determine the corresponding cumulative probability.
Step 1: Calculate the z-score.
The z-score formula is given by:
z = (x - μ) / σ
Where:
x = the value we want to convert to a z-score (in this case, 5 inches)
μ = the mean of the distribution (65 inches)
σ = the standard deviation of the distribution (3.5 inches)
Plugging in the values, we get:
z = (5 - 65) / 3.5
z = -60 / 3.5
z ≈ -17.143
Step 2: Find the cumulative probability.
Using a standard normal distribution table or calculator, we can find the cumulative probability corresponding to the z-score of -17.143. This represents the proportion of the distribution below the value of 5 inches.
Looking up the z-score in the table, we find that the cumulative probability is practically 0.
Step 3: Calculate the number of students.
The cumulative probability of 0 means that none of the 200 students in this sample have a height less than 5 inches.