Question

A health psychologist hypothesizes that depression levels in the nation have been decreasing over the years. In the late 2000s, the average score on the Birk Depression Inventory (BDI) in the nation was 17 with a variance of 56.25. A current sample of 16 generates an average BDI score of 13.4. What can be concluded with an α of 0.10?

c) Obtain/compute the appropriate values to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value_________ =  ; test statistic = __________
Decision: Reject H0 or Fail to reject H0

d) If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[ _________ , ________ ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and select "na" below.
d = ______ ;   na trivial effect small effect medium effect large effect
r² = ______ ;   na trivial effect small effect medium effect large effect

f) Make an interpretation based on the results.

The current population is significantly more depressed.

The current population is significant less depressed.

   The current population does not significantly differ in depression

Answer 1

A health psychologist hypothesizes that depression levels in the nation have been decreasing over the years.

The average score on the Birk Depression Inventory (BDI) in the nation was 17 with a variance of 56.25.

So

= 17 , Variance 2 = 56.25

So Standard deviation would be

= = 7.5

A current sample of 16 generates an average BDI score of 13.4.

Thus n = 16

Null and alternative hypotheses are

To Test

H0 : 17    { depression levels in the nation have been decreasing over the years }

vs

H1 : > 17 {depression levels in the nation have not been decreasing significantly over the years}

C) To obtaine / compute the appropriate values to make a decision about H0.

Critical value :- Now we have information about population variance 2 (2 = 56.25) , so we can use z-scores as critical value

{ Note that if population variance is unkown then we use t-score with n-1 degree of freedom }

Since this is one tail test as alternative hypothesis is of " > " type .

At significance level α of 0.10 z-critical value is = 1.28

Critical Value = 1.28

Test Statistics TS :

TS =

Here = 13.4 , = 17   ,    = 7.5 , n =16

So

TS =

TS =

TS = -1.92

Thus calculates value of test statistics is TS = -1.92

So we have

critical value    1.28 ; test statistic =    -1.92 _

Decision:- Since Since this is one tail test as alternative hypothesis is of " > " type , so we reject null hypothesis is calculated test statistics is greater than critical value .

Here TS = -1.92 < 1.28 ( i.e critical value )

Sicen test statistics value is less than critical value we fail to reject null hypothesis . So we can conclude that depression levels in the nation have been decreasing over the years

Decision: Fail to reject H0

d) If appropriate, compute the CI.

Note that : - It is not mentioned weather confidence interval for two sided or one sided is required and signigicance level is not mentioned .

So we will use information from part c) where we have one-tail test and = 0.10 { i.e 90% confidence }.

So here we make 90% confidence interval for one -tail test where alternative hypothesis is of " > " type.

Confidence interval for meanaverage score if given by

CI = ( - , + * )

Now mean = 17 , = 1.28 { for =.10 } ,    = 7.5 , n =16

CI = ( - , + * )

CI = ( - , 17 + 1.28 * ) = ( , 17 + 2.4 )

CI = ( - , 19.4 )

So 90% confidence interval for one sided average score is = ( - , 19.4 )

Here 90% confidence interval which is (- , 19.4 ) means that we do not reject null hypothesis H0 for any score/number ( score on the Birk Depression Inventory ) less than 19.4 at significance level α = 0.10.

So 19.4 is upper limit and any score for depression above 19.4 we reject null hypotheiss

e) Compute the corresponding effect size(s) and indicate magnitude(s).

Formual and calculation .

Effective size d = = = 0.48

since d = 0.48 0.5 so we can say medium effect

r2 =

Here d2 = (0.48)2 = 0.2304

And df is degree of freedom df = n-1 = 16-1 = 15

So

r2 = = = 0.01512764

d = 0.48

r2 = 0.01512764

f) Make an interpretation based on the results.

since we fail to reject null hypothesis , and concluded that depression levels in the nation have been decreasing over the years .

So the correct option would be

option ii) The current population is significant less depressed.