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.
1)How many students would you expect to have weight between 162 pounds and 178 pounds?
2)What is the probability that a student picked at random weighs less than 165 pounds?
Use process from previous post.
To answer these questions, we need to use the standard normal distribution, also known as the Z-distribution. This distribution allows us to calculate probabilities and find the number of standard deviations a given value is from the mean.
The formula to convert a value from the original distribution to the standard normal distribution is:
Z = (X - μ) / σ
Where:
- Z is the Z-score
- X is the value from the original distribution
- μ is the mean of the original distribution
- σ is the standard deviation of the original distribution
Now, let's apply these steps to answer the questions:
1) How many students would you expect to have weight between 162 pounds and 178 pounds?
First, we need to calculate the Z-scores for the lower and upper bounds using the formula mentioned above:
Z_l = (162 - 170) / 5 = -1.6
Z_u = (178 - 170) / 5 = 1.6
Next, we need to find the corresponding probabilities using the Z-score table or a calculator. The table or calculator will give us the area under the curve for the Z-scores.
P(162 < X < 178) = P(Z_l < Z < Z_u)
Using the Z-table or calculator, we find the probability for Z_l = -1.6 as 0.0548 and the probability for Z_u = 1.6 as 0.9452.
Now, we take the difference of these two probabilities to find the probability of having weight between 162 and 178 pounds:
P(162 < X < 178) = 0.9452 - 0.0548 = 0.8904
To find the number of students, we multiply this probability by the total number of students:
Number of students = Probability * Total number of students
= 0.8904 * 100
= 89.04
Therefore, we would expect approximately 89 students to have weights between 162 and 178 pounds.
2) What is the probability that a student picked at random weighs less than 165 pounds?
First, we calculate the Z-score for 165 pounds using the formula mentioned earlier:
Z = (165 - 170) / 5 = -1
We want to find the probability P(X < 165), which can be calculated by finding the area to the left of the Z-score in the Z-table or using a calculator.
P(X < 165) = P(Z < -1)
Using the Z-table or calculator, we find the probability for Z = -1 as 0.1587.
Therefore, the probability that a randomly selected student weighs less than 165 pounds is 0.1587, or 15.87%.
By following these steps and using the Z-distribution, we can answer questions related to probabilities and estimates in a normally distributed dataset.