Question

In New York State, the mean salary for high school teachers in 2017 was $103,010 with a standard deviation of $9,640. Only Alaska’s mean salary was higher! Assume New York’s state salaries follow a normal distribution.

a. What percent of New York’s state high school teachers earn between $88,000 and $93,000? (Round intermediate calculations to 2 decimal places and final answer to 2 decimal places.)

b. What percent of New York’s state high school teachers earn between $93,000 and $108,000? (Round intermediate calculations to 2 decimal places and final answer to 2 decimal places.)

c. What percent of New York’s state high school teachers earn less than $78,000? (Round intermediate calculations to 2 decimal places and final answer to 2 decimal places.)

Answer 1

To solve these questions, we need to use the Z-score formula and the standard normal distribution table. The Z-score formula is:

Z = (X - μ) / σ

Where:

Z is the standard score (Z-score).

X is the raw score.

μ is the mean of the distribution.

σ is the standard deviation of the distribution.

a. To find the percent of New York's state high school teachers earning between $88,000 and $93,000, we first need to calculate the Z-scores for both values.

Z1 = (88,000 - 103,010) / 9,640 Z2 = (93,000 - 103,010) / 9,640

Next, we look up the cumulative probability for each Z-score in the standard normal distribution table. Subtracting the cumulative probability of Z1 from Z2 gives us the percentage of teachers earning between $88,000 and $93,000.

P1 = Lookup(Z1) P2 = Lookup(Z2)

Percentage = P2 - P1

b. Similarly, to find the percent of New York's state high school teachers earning between $93,000 and $108,000, we calculate the Z-scores for both values.

Z3 = (93,000 - 103,010) / 9,640 Z4 = (108,000 - 103,010) / 9,640

Then, we use the cumulative probabilities associated with Z3 and Z4 to calculate the percentage.

P3 = Lookup(Z3) P4 = Lookup(Z4)

Percentage = P4 - P3

c. To find the percent of New York's state high school teachers earning less than $78,000, we calculate the Z-score for this value.

Z5 = (78,000 - 103,010) / 9,640

Using the cumulative probability associated with Z5, we can determine the percentage.

P5 = Lookup(Z5)

Now we can calculate the final answer for each question. Note that the "Lookup" function refers to consulting the standard normal distribution table to find the cumulative probability corresponding to a specific Z-score.