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A guidance counselor at a university is investigating demand for study abroad. The question before her is how engineering and humanities majors compare regarding interest in study abroad during the summer. Random samples of 20 engineering and 20 humanities majors each were interviewed. Eight engineering majors and 12 humanities majors expressed interest in study abroad during the summer.

The estimated difference in the proportion of engineering and humanities majors, [tex]$p_E - p_H$[/tex], where [tex]$p_E$[/tex] is the proportion of engineering majors interested in study abroad and [tex][tex]$p_H$[/tex][/tex] is the proportion for humanities majors, has which requirement for Normality?

A. [tex]$x_1, x_2 \geq 10$[/tex] and [tex]$n_1 - x_1, n_2 - x_2 \geq 10$[/tex]
B. [tex][tex]$n_1 + n_2 \geq 60$[/tex][/tex]
C. [tex]$x_1, x_2 \geq 30$[/tex] and [tex]$n_1 - x_1, n_2 - x_2 \geq 30$[/tex]
D. [tex][tex]$n_1 \geq 30$[/tex][/tex] and [tex]$n_2 \geq 30$[/tex]



Answer :

GinnyAnswer
Certainly! Let's break down the provided information step-by-step to determine the requirements for Normality in estimating the difference between the proportions of engineering and humanities majors interested in study abroad.

We have the following details:
- [tex]\( n_1 = 20 \)[/tex]: number of engineering majors interviewed.
- [tex]\( n_2 = 20 \)[/tex]: number of humanities majors interviewed.
- [tex]\( x_1 = 8 \)[/tex]: number of engineering majors interested in study abroad.
- [tex]\( x_2 = 12 \)[/tex]: number of humanities majors interested in study abroad.

From this, let's evaluate each requirement for Normality separately:

1. [tex]\( x_1, x_2 \geq 10 \)[/tex] and [tex]\( n_1 - x_1, n_2 - x_2 \geq 10 \)[/tex]
- [tex]\( x_1 = 8 \)[/tex] (which is less than 10)
- [tex]\( x_2 = 12 \)[/tex] (greater than or equal to 10)
- [tex]\( n_1 - x_1 = 20 - 8 = 12 \)[/tex] (greater than or equal to 10)
- [tex]\( n_2 - x_2 = 20 - 12 = 8 \)[/tex] (which is less than 10)
- Therefore, this condition is not met.

2. [tex]\( n_1 + n_2 \geq 60 \)[/tex]
- [tex]\( n_1 + n_2 = 20 + 20 = 40 \)[/tex] (which is less than 60)
- Therefore, this condition is not met.

3. [tex]\( x_1, x_2 \geq 30 \)[/tex] and [tex]\( n_1 - x_1, n_2 - x_2 \geq 30 \)[/tex]
- [tex]\( x_1 = 8 \)[/tex] (which is less than 30)
- [tex]\( x_2 = 12 \)[/tex] (which is less than 30)
- [tex]\( n_1 - x_1 = 20 - 8 = 12 \)[/tex] (which is less than 30)
- [tex]\( n_2 - x_2 = 20 - 12 = 8 \)[/tex] (which is less than 30)
- Therefore, this condition is not met.

4. [tex]\( n_1 \geq 30 \)[/tex] and [tex]\( n_2 \geq 30 \)[/tex]
- [tex]\( n_1 = 20 \)[/tex] (which is less than 30)
- [tex]\( n_2 = 20 \)[/tex] (which is less than 30)
- Therefore, this condition is not met.

In summary, none of the requirements for Normality are met given the counts and proportions for the engineering and humanities majors provided in the data. Thus, the results for the normality check are all negative.

The answers are:
1. [tex]\( x_1, x_2 \geq 10 \)[/tex] and [tex]\( n_1 - x_1, n_2 - x_2 \geq 10 \)[/tex]: False
2. [tex]\( n_1 + n_2 \geq 60 \)[/tex]: False
3. [tex]\( x_1, x_2 \geq 30 \)[/tex] and [tex]\( n_1 - x_1, n_2 - x_2 \geq 30 \)[/tex]: False
4. [tex]\( n_1 \geq 30 \)[/tex] and [tex]\( n_2 \geq 30 \)[/tex]: False

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