Question

Nineteen immigrants to the U.S. were asked how many years, to the nearest year, they have lived in the U.S. The data are as follows: 2; 5; 7; 2; 2; 10; 20; 15; 0; 7; 0; 20; 5; 12; 15; 12; 4; 5; 10.
Data Frequency Relative frequency Cumulative relative frequency
0 2 2/19 0.1053
2 3 3/19 0.2632
4 1 1/19 0.3158
5 3 3/19 0.4737
7 2 2/19 0.5789
10 2 2/19 0.6842
12 2 2/19 0.7895
15 1 1/19 0.8421
20 1 1/19 1.000

(a) Fix the error in the table. explain how someone might have arrived at the incorrect number(s).
(b) Explain what is wrong with the statement "47% of the people surveyed have lived in the U.S. for 5 years"

Answer 1

Answer:

a) The frequency of the data "15" and "20" is 2 for both, not 1; this means their relative frequency is 2/19 for both, not 1/19; finally, the cumulative relative frequency in the row of the data "15" should be 0.8947, not 0.8421. This error might have happened because someone didn't count the numbers correctly, so they only noticed one "15" and one "20" when, in fact, there were two people that had lived in the U.S. for 15 years, and two more people for 20 years. On the other hand, the error in the cumulative relative frequency happened because it accounted for only one person living in the U.S. for 15 years, instead of two people.

b) Roughly 47% of the people surveyed have lived in the U.S. from 0 to 5 years, not for 5 years. The cumulative relative frequency in this row (47%) accounts for every data gathered so far, not just the "5 years" row. The correct statement would be that 3 out of 19, or 15.8% (relative frequency) of the people surveyed have lived in the U.S. for 5 years.

Step-by-step explanation:

1) First of all, to avoid errors like the one in the problem's table, we should first place the given numbers from least to greatest, so we can construct a new frequency table by ourselves. Let's do just that, and we'll end up with something like this:

0 , 0, 2 , 2, 2, 4, 5, 5, 5, 7, 7, 10, 10, 12, 12 , 15, 15, 20, 20

Now we'll have a much easier time from now on.

2) The second step is to construct the Data and Frequency columns. Just place each unique integer in a new row of the Data column, then count how many times that unique integer was found, and, finally, place that number below the Frequency column (Please refer to the Excel Worksheet provided as an attachment).

Let's do it as follows:

Data     Frequency

0            2

2            3

4            1

5            3

7            2

10           2

12           2

15           2

20          2

Note that we counted "15" and "20" twice! So each one of those rows have a frequency of 2, not 1 as the table presented in the problem suggests.

3) Next, we want to construct the Relative frequency and Cumulative relative frequency columns. For the relative frequency column, we just divide the frequency of each row by the total number of immigrants surveyed, which is 19. For the cumulative relative frequency column, we will get each row's relative frequency, and add the cumulative relative frequency of the row before it. Note that for the first row, the cumulative relative frequency is the same as its relative frequency.

We should get something like this:

Data    Frequency    Relative frequency    Cumulative relative frequency

0            2                   2/19                               0.1053

2            3                   3/19                               0.2632

4            1                    1/19                                0.3158

5            3                   3/19                               0.4737

7            2                   2/19                               0.5789

10           2                   2/19                               0.6842

12           2                   2/19                               0.7895

15           2                   2/19                               0.8947

20          2                   2/19                               1.0000

Note that the relative frequency for both "15" and "20" is 2/19 instead of 1/19! Also, we got a cumulative relative frequency of 0.8947 in the row of "15", instead of 0.8421.

4) a) We have just fixed the error in the table, but we have to explain how someone might have arrived at the incorrect number(s). The most logical way that someone might have gotten the incorrect frequencies of "15" and "20" is that they didn't count the numbers correctly while building the Frequency column. This could have happened because that person probably didn't order the numbers from least to greatest, as we did in Step 1, which makes it way easier to get the frequency of each data without making a mistake.

5) b) We have now to explain what is wrong with the statement "47% of the people surveyed have lived in the U.S. for 5 years.

To answer that, we can refer to the relative frequency of the row of the data "5", which tells us that 3 out of 19 (or roughly 15.8%) of the people surveyed have lived in the U.S. for 5 years. Relative frequency is telling us the percentage of people that have lived for this amount of time.

By contrast, the cumulative relative frequency of this same row tells us that 0.4737, or roughly 47%, of the people surveyed have lived for 5 years or less. Cumulative relative frequency accounts for the data presented in its row, plus the data presented in the rows before it.

So the correct statement would be either:

• 15.8% of the people surveyed have lived in the U.S. for 5 years, or
• Roughly 47% of the people surveyed have lived in the U.S. for 5 years or less.

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