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

Answer:

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

(2.13264 , 2.58736)

Step-by-step 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

$t_{\frac{0.05}{2} , 49} = 2.0096$

Step(ii):-

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

$(x^{-} - t_{(\frac{\alpha }{2},df )} \frac{S}{\sqrt{n} } , (x^{-} +t_{(\frac{\alpha }{2},df )} \frac{S}{\sqrt{n} })$

$(2.36 - 2.0096 \frac{0.8}{\sqrt{50} } , 2.36 +2.0096\frac{0.8}{\sqrt{50} })$

( 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)