1. Suppose you administered an anxiety test to a large sample of people and obtained normally distributed scores with a mean of 45 and a standard deviation of 4. Do not use the web calculator to answer the following questions. Instead, use the Z distribution table below and Appendix A in the Jackson text.
a. If Andrew scored 45 on this test, what is his Z score? Z score = Zero
45-45/4 = 0
b. If Anna scored 30 on this test, what is her Z score? Z score = -3.750
30-45/4 =-3.750
c. If Bill’s Z score was 1.5, what is his real score on this test? His real score is 93.3
1.5 is the Z score the area between the mean and z (0.433+ 0.50 = 0.933 X 100)
There are 200 students in a sample. How many of these students will have scores that fall under the score of 41
Z = (score-mean)/SD = (45-45)/4 = 0
Need the parenthesis.
b. Z = (30-45)/4 = -15/4 = ?
c. No. 1.5 = (x-45)/4
Solve for x.
d. Z = (41-45)4 = -1
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Multiply by 200.
To calculate how many students will have scores that fall under the score of 41, we need to find the proportion or percentage of students with scores less than 41, and then multiply it by the total number of students in the sample (200 in this case).
To find the proportion or percentage, we can use the Z-score formula:
Z = (X - mean) / standard deviation
where X is the score we want to find the proportion for, mean is the mean of the distribution, and standard deviation is the standard deviation of the distribution.
Let's calculate the Z-score for the score of 41:
Z = (41 - 45) / 4
Z = -1
Next, we need to determine the proportion (or percentage) of scores less than -1. We can find this information from the Z-score table or using a calculator or software that provides cumulative probabilities for the standard normal distribution.
Looking up the Z-score of -1 in the table, we find the proportion/percentage to be approximately 0.1587 (or 15.87%).
Now, we can calculate the number of students with scores less than 41 by multiplying this proportion by the total number of students in the sample:
Number of students with scores < 41 = 0.1587 * 200
Number of students with scores < 41 = 31.74
Therefore, approximately 31.74 students out of the 200 students in the sample will have scores that fall under the score of 41.
To determine the number of students that will have scores below 41, we need to calculate the proportion of the distribution that falls below the score of 41.
To do this, we need to find the corresponding Z-score for 41 using the formula:
Z = (X - μ) / σ
where:
X = score
μ = mean
σ = standard deviation
In this case, X = 41, μ = 45, σ = 4. Plugging these values into the formula, we get:
Z = (41 - 45) / 4
Z = -4 / 4
Z = -1
Now, we can use the Z-distribution table to find the proportion of the distribution that is below a Z-score of -1.
Looking at the table, we find that the area to the left of -1 is approximately 0.1587.
To find the number of students in the sample with scores below 41, we multiply this proportion by the total number of students in the sample:
Number of students = Proportion * Total number of students
Number of students = 0.1587 * 200
Number of students ≈ 31.74
Therefore, approximately 32 students in the sample will have scores below 41.