Suppose the people living in a city have a mean score of 48 and a standard deviation score of 7 on a measure of concerns about the environment.
Assume that these concerns are normally distributed using the 50%- 34% - 14% figures.
Appoxatmaily what percentage of people have a score
A. Above 48
B. Above55
C. Above 34
D. Above 41
E. Below 48
F. Below 55
G. Below 34
H. Below 41
f,e,g,or h
Another method.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
To answer these questions, we can use the Z-score formula:
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the given value (score),
μ is the mean score,
σ is the standard deviation.
A. Above 48:
Z = (48 - 48) / 7 = 0
Using the 50% - 34% - 14% rule, we know that the area to the right of 0 on the standard normal distribution is 50%. Therefore, approximately 50% of people have a score above 48.
B. Above 55:
Z = (55 - 48) / 7 ≈ 1
The area to the right of 1 on the standard normal distribution is approximately 34%. Hence, approximately 34% of people have a score above 55.
C. Above 34:
Z = (34 - 48) / 7 ≈ -2
The area to the right of -2 on the standard normal distribution is approximately 97.72%. So approximately 97.72% of people have a score above 34.
D. Above 41:
Z = (41 - 48) / 7 ≈ -1
The area to the right of -1 on the standard normal distribution is approximately 84.13%. Therefore, approximately 84.13% of people have a score above 41.
E. Below 48:
Since 48 falls on the mean, we can say that approximately 50% of people have a score below 48.
F. Below 55:
Z = (55 - 48) / 7 ≈ 1
The area to the left of 1 on the standard normal distribution is approximately 84.13%. So approximately 84.13% of people have a score below 55.
G. Below 34:
Z = (34 - 48) / 7 ≈ -2
The area to the left of -2 on the standard normal distribution is approximately 2.28%. Hence, approximately 2.28% of people have a score below 34.
H. Below 41:
Z = (41 - 48) / 7 ≈ -1
The area to the left of -1 on the standard normal distribution is approximately 15.87%. Therefore, approximately 15.87% of people have a score below 41.
To answer these questions, we need to use the concept of z-scores, which measure the number of standard deviations a particular value is from the mean.
First, we will calculate the z-scores for the given values using the formula:
z = (x - mean) / standard deviation
where x is the given value, mean is the mean score, and standard deviation is the standard deviation score.
A. To find the percentage of people with a score above 48, we need to calculate the z-score for 48 and then find the percentage of the population above that z-score. Since the mean score is 48 and the question is asking for values above 48, the z-score for this value would be 0.
Percentage above 48 = (100 - 50) / 2 = 25%
B. To find the percentage of people with a score above 55, we calculate the z-score for 55 and then find the percentage of the population above that z-score.
z = (55 - 48) / 7 = 1
Percentage above 55 = 50 - 34 = 16%
C. To find the percentage of people with a score above 34, we calculate the z-score for 34 and then find the percentage of the population above that z-score.
z = (34 - 48) / 7 = -2
Percentage above 34 = 50 + 34 = 84%
D. To find the percentage of people with a score above 41, we calculate the z-score for 41 and then find the percentage of the population above that z-score.
z = (41 - 48) / 7 = -1
Percentage above 41 = 50 + 34 + 14 = 98%
E. To find the percentage of people with a score below 48, we use the percentage above 48 (which we calculated in part A) and subtract it from 100%.
Percentage below 48 = 100% - Percentage above 48 = 100% - 25% = 75%
F. To find the percentage of people with a score below 55, we use the percentage above 55 (which we calculated in part B) and subtract it from 100%.
Percentage below 55 = 100% - Percentage above 55 = 100% - 16% = 84%
G. To find the percentage of people with a score below 34, we use the percentage above 34 (which we calculated in part C) and subtract it from 100%.
Percentage below 34 = 100% - Percentage above 34 = 100% - 84% = 16%
H. To find the percentage of people with a score below 41, we use the percentage above 41 (which we calculated in part D) and subtract it from 100%.
Percentage below 41 = 100% - Percentage above 41 = 100% - 98% = 2%
So, the percentages can be summarized as follows:
A. Above 48: 25%
B. Above 55: 16%
C. Above 34: 84%
D. Above 41: 98%
E. Below 48: 75%
F. Below 55: 84%
G. Below 34: 16%
H. Below 41: 2%