The weights of the 100 students in an introductory statistics class are normally distributed, with a mean of 170 pounds and a standard deviation of 5 pounds.

How many students would you expect to have weight between 162 pounds and 178 pounds?

Use same process as in previous post.

To determine the number of students that would be expected to have weights between 162 pounds and 178 pounds, we need to use the properties of a normal distribution curve.

Step 1: Find the z-scores
A z-score measures how many standard deviations an individual value is from the mean of the distribution.
For the lower bound of 162 pounds:
z1 = (162 - 170) / 5 = -1.6
For the upper bound of 178 pounds:
z2 = (178 - 170) / 5 = 1.6

Step 2: Determine the area under the curve
To find the area under the normal distribution curve between these two z-scores, we can use a standard normal distribution table or a calculator.

Alternatively, we can use the cumulative distribution function (CDF) of the standard normal distribution to find the cumulative probability between the two z-scores.
P(z1 < Z < z2) = P(Z < z2) - P(Z < z1)

Step 3: Calculate the expected number of students
The cumulative probability P(z1 < Z < z2) represents the proportion of the population that falls between the lower and upper bounds. To calculate the expected number of students within this range, we can multiply this proportion by the total number of students in the class.

Expected number of students = P(z1 < Z < z2) * Total number of students

So, to find the expected number of students with weights between 162 pounds and 178 pounds, follow these steps:

1. Calculate the z-scores:
z1 = (162 - 170) / 5 = -1.6
z2 = (178 - 170) / 5 = 1.6

2. Use a standard normal distribution table or calculator to find the cumulative probability between z1 and z2:
P(z1 < Z < z2) = P(Z < z2) - P(Z < z1)

3. Multiply the cumulative probability by the total number of students:
Expected number of students = P(z1 < Z < z2) * Total number of students.

Please note that the expected number of students will be an approximation based on the assumption that the weights of the students follow a normal distribution.