Question

The Acculturation Rating Scale for Mexican Americans (ARSMA) is a psychological testdeveloped to measure the degree of Mexican/Spanish versus Anglo/Englishacculturation of Mexican Americans. The distribution of ARSMA scores in a populationused to develop the test was approximately normal, with mean 3.0 and standarddeviation 0.8. A further study gave ARSMA to 50 first-generation Mexican Americans.The mean of their scores is x bar = 2.36. If the standard deviation for the first-generationpopulation is also 0.8.

Required:
Give a 95% confidence interval for the mean ARSMA score forfirst-generation Mexican Americans

Answer 1

95% confidence interval for the mean ARSMA score for first-generation Mexican Americans

(2.13264 , 2.58736)

Explanation:

Step(i):-

Mean of the Population = 3.0

Standard deviation of the Population = 0.8

Given Mean of the sample(x⁻ ) = 2.36

Standard deviation of the sample (S) = 0.8

size of the sample = 50

Level of significance =0.05

Degrees of freedom = n-1 = 50-1 = 49

`t_{(0.05)/(2) , 49} = 2.0096`

Step(ii):-

95% confidence interval for the mean ARSMA score for first-generation Mexican Americans

`(x^(-) - t_{((\alpha )/(2),df )} (S)/(√(n) ) , (x^(-) +t_{((\alpha )/(2),df )} (S)/(√(n) ))`

`(2.36 - 2.0096 (0.8)/(√(50) ) , 2.36 +2.0096(0.8)/(√(50) ))`

( 2.36 - 0.22736 , 2.36 + 0.22736)

(2.13264 , 2.58736)

Final answer:-

95% confidence interval for the mean ARSMA score for first-generation Mexican Americans

(2.13264 , 2.58736)