The probability that a randomly selected education major received a starting salary offer greater than $52,350 is . The probabil

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The probability that a randomly selected education major received a starting salary offer greater than $52,350 is . The probabil

ity that a randomly selected education major received a starting salary offer between $45,000 and $52,350 is . (Hint: The standard normal distribution is perfectly symmetrical about the mean, the area under the curve to the left (and right) of the mean is 0.5. Therefore, the area under the curve between the mean and a z-score is computed by subtracting the area to the left (or right) of the z-score from 0.5.) What percentage of education majors received a starting offer between $38,500 and $45,000? 6.68%

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1 answer:

Free_Kalibri [3.7K] · Answer 1 · 2026-04-08T06:13:08+03:00

Answer:

1. 0.0808

2. 0.5371

3. 58.2%

Explanation:

The first part of the question, with the input data, is missing.

This is missing part:

We assume that X, the initial salary offer for education majors is normally distributed with a mean of $46,292 and a standard deviation of $4,320

Solution

In the first question: The probability that a randomly selected education major had a starting salary offer exceeding $52,350 is ________

We need to calculate the Z-score.

The Z-score reflects the standardized value of the random variable, indicating how many standard deviations a value is from the mean:

$Z-score=\dfrac{X-\mu}{\sigma}$

$Z-score=\dfrac{\$52,350-\$46,292}{\$4,320}=1.40$

$P(X>\$52,350)=P(Z>1.40)=P(Z$

The standard normal distribution tables provide cumulative probabilities as the area under the bell-shaped curve. P(Z>1.40) indicates the area to the right of Z = 1.40, alternatively, P(Z<-1.40) represents the area to the left of Z = -1.40.

According to the table, P(Z>1.40) = 0.0808, equivalent to 8.08%

In the second question: The probability that a randomly selected education major received a starting salary offer between $45,000 and $52,350 is _______.

Now we must determine the area under the curve connecting the two Z-scores.

$Z(X=$45,000)=\dfrac{\$45,000-\$46,292}{\$4,320}=-0.30$

We seek the area between Z = -0.30 and Z = 1.40.

This is done via P(Z<1.40) - P(Z < - 0.30) = P(Z > - 1.40) - P(Z < - 0.30)

= 1 - P(Z<-1.40) - P( Z < -0.30)

Now, we need to work with the area under the respective curves relating to Z = - 1.40 and to the left of Z = - 0.30.

From the corresponding table, this evaluation yields: 1 - 0.0808 - 0.3821 = 0.5371

Lastly, for question three: What percentage of education majors received a starting offer between $38,500 and $45,000?

For X = $38,500:

$Z=\dfrac{\$38,500-\$46,292}{\$4,320}=-1.80$

For X = $45,000

Z = -0.3 (calculated above)

Next, we need to evaluate the area under the curve to the right of Z = - 1.80 and to the left of Z = - 0.3

This results in P (Z > - 1.80) - P (Z < - 0.3)

1 - P (Z < - 1.80) - P (Z < - 0.3)

1 - 0.0359 - 0.3821 = 0.582 = 58.2%