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:

Table $1.35$was produced.

a. Fix the errors in Table $1.35$. Also, explain how someone might have arrived at the incorrect number(s).

b. Explain what is wrong with this statement: “$47$percent of the people surveyed have lived in the U.S. for $5$years.”

c. Fix the statement in b to make it correct.

d. What fraction of the people surveyed have lived in the U.S. five or seven years?

e. What fraction of the people surveyed have lived in the U.S. at most $12$years?

f. What fraction of the people surveyed have lived in the U.S. fewer than $12$years?

g. What fraction of the people surveyed have lived in the U.S. from five to $20$years, inclusive?

Short answer

(a)

Data	Frequency	Relative Frequency	Cumulative Relative Frequency
0	2	$\frac{2}{19}$	0.1053
2	3	$\frac{3}{19}$	0.2632
4	1	$\frac{1}{19}$	0.3158
5	3	$\frac{3}{19}$	0.4737
7	2	$\frac{2}{19}$	0.5789
10	2	$\frac{2}{19}$	0.6842
12	2	$\frac{2}{19}$	0.7895
15	2	$\frac{2}{19}$	0.8947
20	2	$\frac{2}{19}$	1

(b)$47\%$is the Cumulative Relative Frequency for 5 years, not the Relative Frequency.

(c) $47\%$of the people surveyed have lived in the U.S. for five years or less.

(d) The fraction of the people surveyed who have lived in the U.S. for five or seven years is $\frac{5}{19}$

(e) The fraction of the people surveyed who have lived in the U.S. at mostlocalid="1647697914496" $12$years is $\frac{14}{19}$

(f) The fraction of the people surveyed who have lived in the U.S. for fewer than $12$years is $\frac{12}{19}$

(g) The fraction of the people surveyed who have lived in the U.S. from five to $20$years is$\frac{13}{19}$

Step-by-step solution

Given information (part a)

Given the table of information about the number of years, the immigrants have lived in the U.S.

Explanation (part a)

The row of $20$has $1$frequency and the row of $15$has $1$frequency, which is not correct. The correct answer is that row $20$has $2$frequencies and the row of $15$has $2$frequencies. The table with the corrected number is given as below:

Data	Frequency	Relative Frequency	Cumulative relative frequency
0	2	$\frac{2}{19}$	0.1053
2	3	$\frac{3}{19}$	0.2632
4	1	$\frac{1}{19}$	0.3158
5	3	$\frac{3}{19}$	0.4737
7	2	$\frac{2}{19}$	0.5789
10	2	$\frac{2}{19}$	0.6842
12	2	$\frac{2}{19}$	0.7895
15	2	$\frac{2}{19}$	0.8947
17	2	$\frac{2}{19}$	1.0000

Given information (part b)

Given the table of information about the number of years, the immigrants have lived in the U.S.

Explanation (part b)

The given statement is wrong because it has been calculated by using the column of cumulative relative frequency which is not correct. So, it is suggested to use the column of relative frequency.

Given information (part c)

Given the table of information about the number of years, the immigrants have lived in the U.S.

Explanation (part c)

The percentage of the surveyed people who lived for five years can be calculated by using the relative frequency column of the above table. So, it will be:

$$\begin{gathered} =\frac{3}{19} \\ =0.1579 \\ =15.79\% \end{gathered}$$

So, the given statement that $47\%$of the people surveyed have lived in the U.S. for five years is wrong.

Given information (part d)

Given the table of information about the number of years, the immigrants have lived in the U.S.

Explanation (part d)

Fraction of the people who have lived in the U.S. five or seven years

$$\begin{gathered} =\text{Fractionfor5years+Fractionfor7years} \\ =\frac{3}{19}+\frac{2}{19} \\ =\frac{5}{19} \end{gathered}$$

Given information (part e)

Given the table of information about the number of years, the immigrants have lived in the U.S.

Explanation (part e)

$$\begin{gathered} \text{Fractionforatmost12years=Sumoffractionbefore12years} \\ =\frac{2}{19}+\frac{3}{19}+\frac{1}{19}+\frac{3}{19}+\frac{2}{19}+\frac{2}{19}+\frac{2}{19} \\ =\frac{14}{19} \end{gathered}$$

Given information (part f)

Given the table of information about the number of years, the immigrants have lived in the U.S.

Explanation (part f)

$$\begin{gathered} \text{Fractionforfewerthan12years=Sumoffractionbefore12years} \\ =\frac{2}{19}+\frac{3}{19}+\frac{1}{19}+\frac{3}{19}+\frac{2}{19}+\frac{2}{19} \\ =\frac{12}{19} \end{gathered}$$

Given information (part g)

Given the table of information about the number of years, the immigrants have lived in the U.S.

Explanation (part g)

$$\begin{gathered} \text{Fractionfor5to20years=Sumoffractionsbetween5and20years} \\ =\frac{3}{19}+\frac{2}{19}+\frac{2}{19}+\frac{2}{19}+\frac{2}{19}+\frac{2}{19} \\ =\frac{13}{19} \end{gathered}$$