Ages of self-employed immigrants. Is self-employment for immigrant workers a faster route to economic advancement in the country? This was one of the questions studied in research published in the International Journal of Manpower (Vol. 32, 2011). One aspect of the study involved comparing the ages of self-employed and wage-earning immigrants. The researcher found that in Sweden, native wage earners tend to be younger than self-employed natives. However, immigrant wage earners tend to be older than self-employed immigrants. This inference was based on the table's summary statistics for male Swedish immigrants.

	Self-employed immigrants	Wage-earning immigrants
Sample Size	$\mathbf{870}$	$\mathbf{84}\mathbf{,}\mathbf{875}$
Mean	$\mathbf{44}\mathbf{.}\mathbf{88}$	$\mathbf{46}\mathbf{.}\mathbf{79}$

Source: Based on L. Andersson, "Occupational Choice and Returns to Self-Employment Among Immigrants," International Journal of Manpower, Vol. 32, No. 8, 2011 (Table I).

a. Based on the information given, why is it impossible to provide a measure of reliability for the inference "Self-employed immigrants are younger, on average, than wage-earning immigrants in Sweden"?

b. What information do you need to measure reliability for the inference, part a?

c. Give a value of the test statistic that would conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants if you are willing to risk a Type I error rate of .01.

d. Assume that s, the standard deviation of the ages is the same for both self-employed and wage-earning immigrants. Give an estimate of s that would lead you to conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants using $\mathbf{\text{α=0.01}}$ .

e. Is the true value of s likely to be larger or smaller than the one you calculated in part d?

Step by step solution

01

Step-by-Step Solution Step 1: (a) State the reason

It is impossible to provide reliability for the inference "Self-employed immigrants are younger, on average, than wage-earning immigrants in Sweden" because the measure of variability or variance of data has not been provided.

02

(b) State the required information

To obtain a measure of reliability for the inference, "Self-employed immigrants are younger, on average, than wage-earning immigrants in Sweden," we need the measure of variability or variance of data.

03

(c) Give a value of the test statistic

Let ${\mathrm{\mu }}_1$ be the true mean age of self-employed immigrants and ${\mathrm{\mu }}_2$ be the true mean age of wage-earning immigrants.

Null Hypothesis: There is no significant difference between the true mean age of self-employed immigrants and the true mean age of wage-earning immigrants.

${\mathrm{H}}_0:{\mathrm{\mu }}_1-{\mathrm{\mu }}_2=0$

Alternate Hypothesis: The true mean age of self-employed immigrants is lesser than the true mean age of wage-earning immigrants.

${\mathrm{H}}_{\mathrm{a}}:{\mathrm{\mu }}_1-{\mathrm{\mu }}_2<0$

The critical value for the left-tailed test $\mathrm{\alpha }=0.01$is $-2.33$

If ${\mathrm{z}}_{\mathrm{data}}\le -2.33$then, reject the null hypothesis.

04

(d) Find the value of the Standard Deviation

$\begin{array}{l}z=\frac{(\overline{x_1}-{\overline{x}}_2)-({\mu }_1-{\mu }_2)}{\sqrt{(\frac{{\sigma }^2}{n_1}+\frac{{\sigma }^2}{n_2})}}\\ \Rightarrow \frac{(44.88-46.79)-0}{\sqrt{\left(\frac{{\sigma }^2}{870}+\frac{{\sigma }^2}{84875}\right)}}\le -2.33\\ \Rightarrow \frac{-1.91}{\sqrt{{\sigma }^2\left(\frac{1}{870}+\frac{1}{84875}\right)}}\le -2.33\\ \begin{array}{l}\Rightarrow \frac{-1.91}{\sigma \sqrt{(\frac{1}{870}+\frac{1}{84875})}}\le -2.33\\ \Rightarrow \frac{-1.91}{\sigma \left(0.034\right)}\le -2.33\\ \Rightarrow \frac{-1.91}{-2.33\left(0.034\right)}\le \sigma \\ \Rightarrow \sigma \le 24.11\end{array}\end{array}$

05

(e) State the conclusion about the standard deviation

The true value $\sigma$ is likely to be smaller than $24.11$. Moreover, the true value $\sigma$is too large for the ages of people.

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