Assume that X, the starting salary offer to psychology majors is normally distributed with a mean of $47,507 and a SD of $5000.
1. The probability that a randomly selected psychology major received a starting salary offer greater than $52,000 is______.
2. The probability a randomly selected psychology majors received starting salary offer between $42,000 ad $52,000 is _____.
3. What percent of psychology majors received a starting offer between $38,000 ad $45,000?
4. Twenty percent of psychology majors were offered a starting salary less than___.
Would you please explain the solution instead of just giving the answer. I'd appreciate it very much, thank you.
1. To calculate the probability that a randomly selected psychology major received a starting salary offer greater than $52,000, you need to find the area under the normal distribution curve to the right of $52,000.
First, you need to standardize the value of $52,000 using the formula z = (x - μ) / σ, where x is the value ($52,000 in this case), μ is the mean ($47,507), and σ is the standard deviation ($5,000).
z = ($52,000 - $47,507) / $5,000
z = $4,493 / $5,000
z = 0.8986
Next, you can use a standard normal distribution table or a calculator to find the probability corresponding to the standardized value of 0.8986. The probability (p) that a randomly selected psychology major received a starting salary offer greater than $52,000 is equal to 1 minus the cumulative probability up to the standardized value.
p = 1 - cumulative probability of Z ≤ 0.8986
2. To find the probability that a randomly selected psychology major received a starting salary offer between $42,000 and $52,000, you need to find the area under the normal distribution curve between these two values.
First, standardize the lower value of $42,000 and the upper value of $52,000 using the same formula as before.
Lower z = ($42,000 - $47,507) / $5,000
Upper z = ($52,000 - $47,507) / $5,000
Next, find the cumulative probability of the lower z-value and the cumulative probability of the upper z-value. The probability (p) that a randomly selected psychology major received a starting salary offer between $42,000 and $52,000 is the difference between these two cumulative probabilities.
p = cumulative probability of Z ≤ Upper z - cumulative probability of Z ≤ Lower z
3. To calculate the percentage of psychology majors who received a starting offer between $38,000 and $45,000, you can use the same method as in question 2.
First, standardize the lower value of $38,000 and the upper value of $45,000 using the formula.
Lower z = ($38,000 - $47,507) / $5,000
Upper z = ($45,000 - $47,507) / $5,000
Next, find the cumulative probability of the lower z-value and the cumulative probability of the upper z-value. The difference between these two cumulative probabilities gives you the probability (p) of psychology majors receiving a starting offer between $38,000 and $45,000. Multiply this probability by 100 to convert it to a percentage.
p = (cumulative probability of Z ≤ Upper z - cumulative probability of Z ≤ Lower z) * 100
4. To determine the starting salary that is less than which 20% of psychology majors were offered, you need to find the z-value corresponding to the cumulative probability of 0.2.
Using a standard normal distribution table or calculator, find the z-value such that the cumulative probability of Z ≤ that z-value is equal to 0.2.
Then, unstandardize the z-value using the formula x = (z * σ) + μ, where z is the z-value, σ is the standard deviation, and μ is the mean.
x = (z * $5,000) + $47,507
This will give you the salary value below which 20% of psychology majors were offered a starting salary.
Certainly! I'll explain how to solve each of these questions step by step.
1. The probability that a randomly selected psychology major received a starting salary offer greater than $52,000:
To find this probability, we need to calculate the area under the normal distribution curve above $52,000. We can calculate this using the standard normal distribution table or by converting the value to a standard score (also known as z-score) and using the standard normal distribution table.
To convert the value $52,000 to a z-score, we use the formula:
z = (x - μ) / σ
Where:
x = $52,000 (the value we want to convert to a z-score)
μ = $47,507 (the mean)
σ = $5,000 (the standard deviation)
Substituting the values in, we get:
z = ($52,000 - $47,507) / $5,000
Once we have the z-score, we can look it up in the standard normal distribution table to find the corresponding probability. The table provides the area to the left of the z-score. Since we want the area to the right of $52,000, we subtract the area from 1.
2. The probability a randomly selected psychology major received a starting salary offer between $42,000 and $52,000:
To find this probability, we need to calculate the area under the normal distribution curve between $42,000 and $52,000. We can use the same approach as in question 1.
First, we find the z-scores for $42,000 and $52,000 using the formula from question 1. Then, we subtract the area to the left of $42,000 from the area to the left of $52,000 to get the probability between the two values.
3. What percent of psychology majors received a starting offer between $38,000 and $45,000:
To find this percent, we again need to calculate the area under the normal distribution curve between $38,000 and $45,000. We use the same approach as in question 2.
Find the z-scores for $38,000 and $45,000, then subtract the area to the left of $38,000 from the area to the left of $45,000. Multiply this probability by 100 to express it as a percentage.
4. Twenty percent of psychology majors were offered a starting salary less than _____:
To find the starting salary less than a certain value, we need to find the corresponding z-score that would give us an area of 20% to the left below that value.
We can use the standard normal distribution table in reverse. Look for the closest probability to 20% and find the corresponding z-score. Then, solve for x using the formula for converting a z-score to a value.
By following these steps, you can calculate the answers to each of the questions.
Twenty percent of information systems majors were offered a starting salary greater than
1) 45000
0.8159
91.92%
72.02