Question

the histogram shows the starting salaries (rounded to the nearest thousand dollars) for college graduates based on a random sample of recent graduates. determine whether the following statement is true or false according to the graph.
if the sample is truly representative, then for a group of 400 college graduates, we can expect about 28 of them to have starting salaries in the $41,000 - $45,000 range
choose the correct answer below.
a. false, because the height of the bar for 41 - 45 is approximately 60 and the total number of graduates is approximately 995: 60/995 is approximately 0.06 but 28/400 is approximately 0.07.
b. false, because the bar for 41 - 45 has the same height as the bar for 66 - 70.
c. false, because the height of the bar for 41 - 45 is approximately 70 and the total number of graduates is approximately 995, 70/995 is approximately 0.07 but 28/400 is approximately 0.06.
d. true, because the height of the bar for 41 - 45 is approximately 70 and the total number of graduates is approximately 995, 70/995 is approximately 0.07 as is 28/400.

Explanation:

Step1: Estimate the total number of graduates in the sample

Looking at the histogram, we can estimate the frequencies for each salary range:

• \(41 - 45\): Approximately \(70\)
• \(46 - 50\): Approximately \(150\)
• \(51 - 55\): Approximately \(300\)
• \(56 - 60\): Approximately \(220\)
• \(61 - 65\): Approximately \(90\)
• \(66 - 70\): Approximately \(40\)
• \(71 - 75\): Approximately \(25\)

The total number of graduates in the sample \(n\approx70 + 150+300 + 220+90 + 40+25=995\)

Step2: Calculate the proportion of graduates in the \(41 - 45\) salary range in the sample

The proportion \(p\) of graduates in the \(41 - 45\) salary range is \(p=\frac{70}{995}\approx0.07\)

Step3: Calculate the expected number of graduates in the \(41 - 45\) salary range for a group of \(400\) graduates

If the sample is representative, the expected number \(E\) of graduates in the \(41 - 45\) salary range for a group of \(N = 400\) graduates is \(E=N\times p\). Substituting \(N = 400\) and \(p\approx0.07\), we get \(E=400\times0.07 = 28\)

Answer:

D. True, because the height of the bar for \(41 - 45\) is approximately \(70\) and the total number of graduates is approximately \(995\), \(70/995\) is approximately \(0.07\) as is \(28/400\)