For a normal population with µ = 100 and σ = 20:
1. What is the probability of a particular score being greater than 110?
2. What is the probability of a particular score being less than 50?
Use z-scores:
z = (x - mean)/sd
Your data:
x = 110, 50
mean = 100
sd = 20
Calculate both z-scores. Use a z-table to determine probability for each score.
I hope this will help get you started.
To calculate the probability of a particular score being greater than or less than a certain value in a normal distribution, we can use the Z-score formula and interpret it using a Z-table.
1. Probability of a particular score being greater than 110:
To calculate this probability, we need to standardize the score using the Z-score formula and then find the corresponding probability in the Z-table.
The Z-score formula is: Z = (X - µ) / σ
Where:
X is the particular score (in this case, 110)
µ is the mean of the population (100)
σ is the standard deviation of the population (20)
Calculating the Z-score:
Z = (110 - 100) / 20
Z = 10 / 20
Z = 0.5
Now, we find the corresponding probability in the Z-table. The Z-table provides the probability of having a Z-score less than a given value. Since we want the probability of a score greater than 110, we need to find the complement of this probability.
Using the Z-table, we find the probability for a Z-score of 0.5: P(Z < 0.5) = 0.6915
However, we need the probability for a Z-score greater than 0.5, so we subtract this value from 1:
P(Z > 0.5) = 1 - P(Z < 0.5) = 1 - 0.6915 = 0.3085
Therefore, the probability of a particular score being greater than 110 is approximately 0.3085 or 30.85%.
2. Probability of a particular score being less than 50:
Again, we will use the Z-score formula to calculate the Z-score and then find the corresponding probability in the Z-table.
Z = (X - µ) / σ
Where:
X is the particular score (in this case, 50)
µ is the mean of the population (100)
σ is the standard deviation of the population (20)
Calculating the Z-score:
Z = (50 - 100) / 20
Z = -50 / 20
Z = -2.5
Now, we find the corresponding probability in the Z-table. The Z-table provides the probability of having a Z-score less than a given value. Since we want the probability of a score less than 50, we directly use the value for P(Z < -2.5) from the Z-table.
Using the Z-table, we find the probability for a Z-score of -2.5: P(Z < -2.5) = 0.0062
Therefore, the probability of a particular score being less than 50 is approximately 0.0062 or 0.62%.