The mean annual starting salary of a new graduate in a certain profession is $43,000 with a standard deviation of $500. What is the probability that the starting salary of a new graduate in this profession will be between $39,500 and $46,500?
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To solve this problem, we need to use the concept of standard deviation and the Z-score.
Step 1: Calculate the Z-score for each boundary value.
The Z-score is a measure of how many standard deviations a particular value is from the mean. It is calculated using the formula: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
For the lower boundary value, X = $39,500, μ = $43,000, and σ = $500:
Z1 = (39500 - 43000) / 500 = -7000 / 500 = -14
For the upper boundary value, X = $46,500:
Z2 = (46500 - 43000) / 500 = 3500 / 500 = 7
Step 2: Look up the Z-values in the Z-table.
The Z-table provides the cumulative probability for each Z-value. We are interested in the probability between these two Z-values.
From the Z-table, the probability of a Z-value being less than -14 is virtually zero (close to 0). The probability of a Z-value being less than 7 is almost 1 (close to 1).
Step 3: Calculate the probability.
To find the probability between the two Z-values, we subtract the smaller probability from the larger probability.
P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)
P(Z1 < Z < Z2) = 1 - 0 = 1
Therefore, the probability that the starting salary of a new graduate in this profession will be between $39,500 and $46,500 is 100% or 1.