Answer:
The well-being for frequent movers is significantly different from well-being in the general population. ( Alternate Hypothesis accepted )
cohen's d = -0.91 , ( Large Effect )
Step-by-step explanation:
Given:-
- A sample of size n = 12
- The population mean u_p = 40
- The sample was taken as:
38, 37, 41, 35, 42, 40, 33, 33, 36, 38, 32, 39
Find:-
On the basis of this sample, is well-being for frequent movers significantly different from well-being in the general population? Use a two-tailed test with α = 0.05.
Solution:-
- State the hypothesis for sample mean u_s is same as population mean u_p.
Null Hypothesis: u_s = 40
Alternate Hypothesis: u_s ≠ 40
- The rejection criteria for the Null hypothesis can be modeled by T-value ( n < 30 ) with significance level α = 0.05.
DOF = n - 1 = 12 - 1 = 11
Significance level α = 0.05
t_α/2 = t_0.025 = +/- 2.201
- For the statistic value we have to compute sample mean u_s given by:
u_s = Σ xi / n
u_s = (38 + 37 + 41 + 35 + 42 + 40 + 33 + 33 + 36 + 38 + 32 + 39) / 12
u_s = 37
- For the statistic value we need population standard deviation S_p given by:
S_p = S_s / √n
Where, S_s : Sample standard deviation.
S_s^2 = Σ (xi - u_s)^2 / (n-1)
=[ 2*(38-37)^2 + (37-37)^2 + (41-37)^2 + (35-37)^2 + (42-37)^2 + (40-37)^2 + 2*(33-37)^2 + (36-37)^2 + (32-37)^2 + (39-37)^2 ] / ( 11 )
S_s^2 = [ 2 + 0 + 16 + 4 + 25 + 9 + 32 + 1 + 25 + 4 ] / 11
S_s^2 = 10.73
S_s = 3.28
The population standard deviation ( S_p ) is:
S_p = 3.28 / √12
S_p = 0.95
- The T-statistics value is computed as follows:
t = ( u_s - u_p ) / S_p
t = ( 37 - 40 ) / 0.95 = -3.16
- Compare the T-statistics (t) with rejection criteria (t_α/2).
-3.16 < -2.201
t < t_α/2 ...... Reject Null Hypothesis.
- The well-being for frequent movers is significantly different from well-being in the general population. ( Alternate Hypothesis accepted )
- The cohen's d is calculated as follows:
cohen's d = ( u_s - u_p ) / S_s
cohen's d = ( 37 - 40 ) / 3.28 = -0.91 , ( Large Effect )
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