Suppose. Population of test-scores is normally distributed with mean 65 and standard deviation 13 x~n (65,13) if you select a test at randome what is the probability that the score is below 60 or that it is between 65 and 70 please show work!
I am reading your question as 65 with a standard deviation of 13. This is a population, we will use z-scores and the normal table.
Below 60
(60 -65)/13 = z-score
You can then use the z-table(normal table) to find the area under the lower tail and this will give you the probability of a score under 60
Between 65 and 70
You have to find both z-scores:
(65-65)/13 = 0 that will be right in the middle of the normal table.
(70-65)/13 = z.
using the table, you have to find the area between 65 and 70.
Since there are different versions of z-tables (depending on the textbook), I can't give you further directions regarding reading your z-table.
To find the probability that the test score is below 60 or between 65 and 70, we need to calculate the area under the normal distribution curve.
Step 1: Standardize the Scores
We start by standardizing the scores using the formula:
Z = (X - μ) / σ
Where:
Z is the standardized score,
X is the observed value,
μ is the mean of the population (65 in this case), and
σ is the standard deviation of the population (13 in this case).
For X = 60:
Z1 = (60 - 65) / 13 = -0.3846
For X = 65:
Z2 = (65 - 65) / 13 = 0
For X = 70:
Z3 = (70 - 65) / 13 = 0.3846
Step 2: Lookup the Standardized Scores
Now, we need to look up the probabilities associated with these standardized scores using a standard normal distribution table or a calculator.
The probability of the score being below 60 corresponds to the area to the left of Z1, and the probability of the score being between 65 and 70 corresponds to the area between Z2 and Z3.
From the standard normal distribution table, the probabilities are as follows:
- For Z1 = -0.3846, the area to the left is 0.35.
- For Z2 = 0, the area between Z2 and Z3 is 0.1537.
- For Z3 = 0.3846, the area to the right is 0.65.
Step 3: Calculate the Total Probability
To find the probability that the score is below 60 or between 65 and 70, we add the individual probabilities together:
P(X < 60 or 65 < X < 70) = P(X < 60) + P(65 < X < 70)
= 0.35 + 0.1537
= 0.5037
So, the probability that the test score is below 60 or between 65 and 70 is approximately 0.5037 or 50.37%.