Assume that the average annual salary for a worker in the United States is $30,500 and that the annual salaries for Americans are normally distributed with a standard deviation equal to $6,250. Find the following:
(A) What percentage of Americans earn below $19,000?
(B) What percentage of Americans earn above $40,000?
This is what I am getting so maybe someone can tell me if I am right and if not where am I going wrong.
Z = (score-mean)/SD
A. Z(19,000)= (19,000-30,500)/6250= -1.84
P(x<19,000)=P(z< -1.84) = 23.40 23%
B. z(40,000) = (40,000-30,500)/6250= 1.52
P(x>40,000) = P (z> 1.52) = + 23.40 23%
To find the percentage of Americans earning below $19,000 and above $40,000, we need to use the standard deviation and average salary given.
(A) To find the percentage of Americans earning below $19,000, we need to calculate the z-score for $19,000 and then find the corresponding area under the normal distribution curve using a z-table.
The z-score formula is: z = (x - μ) / σ
Where:
- x is the value you want to find the z-score for (in this case, $19,000)
- μ is the mean (average) salary ($30,500)
- σ is the standard deviation ($6,250)
Plugging in the values:
z = (19000 - 30500) / 6250
z = -11 / 6250
From the z-score table, we can find that the corresponding area to a z-score of -1.76 is 0.0392. However, since we're looking for the percentage below $19,000, we need to subtract this area from 0.5 (to account for both sides of the normal distribution curve).
Percentage below $19,000 = (0.5 - 0.0392) * 100 = 46.08%
Therefore, approximately 46.08% of Americans earn below $19,000.
(B) To find the percentage of Americans earning above $40,000, we'll use the same z-score formula as in part (A).
z = (40000 - 30500) / 6250
z = 9.6 / 6250
From the z-score table, we can find that the corresponding area to a z-score of 1.54 is 0.9382. To find the percentage above $40,000, we need to subtract this area from 0.5.
Percentage above $40,000 = (0.5 - 0.9382) * 100 = -43.82%
Since we can't have a negative percentage, it means that 43.82% of Americans earn below $40,000.
Therefore, approximately 100% - 43.82% = 56.18% of Americans earn above $40,000.