Question

• 4. A research study was conducted to examine the differences between older and younger adults on perceived life satisfaction. A pilot study was conducted to examine this hypothesis. Ten older adults (over the age of 70) and ten younger adults (between 20 and 30) were give a life satisfaction test (known to have high reliability and validity). Scores on the measure range from 0 to 60 with high scores indicative of high life satisfaction; low scores indicative of low life satisfaction. The data are presented below. The standard error of the difference is 3.85.

1) Calculate the mean, median, mode and standard deviation for each group.

2) Go through the steps for and conduct a t-test. Make a decision regarding the null and state your conclusion.

Older Adults	Younger Adults
45	34
38	22
52	15
48	27
25	37
39	41
51	24
46	19
55	26
46	36

Answer 1

mean, mode, median ,SD

1)for olders data

Mean ˉx=∑x/n

=(45+38+52+48+25+39+51+46+55+46)/10

=445/10

=44.5

Mode :
In the given data, the observation 46 occurs maximum number of times (2)

∴Z=46

madian:25,38,39,45,46,48,51,52,55,

madian =n/2 th obs

=10/2

=5th obsrvation

=46

Variacne:

=[∑dx^2-(∑dx)^2/n ) ] /n-1

=[681-((-5)^2/10) ]/9

=681-2.5/9

=678.5/9

=75.39

SD=sqrt(Var)

SD=8.6827

2)Younger Adults

Mean ˉx=∑x/n

=(34+22+15+27+37+41+24+19+26+36)/10

=281/10

=28.1

Mode :
In the given data, no observation occurs more than once.
Hence the mode of the observations does not exist, means mode=0.

Median :
Observations in the ascending order are :
15,19,22,24,26,27,34,36,37,41

Here, n=10 is even.

M=(Value of(n/2)th observation+Value of(n/2+1)th observation) /2

=(Value of(10/2)th observation+Value of(10/2+1)th observation) / 2

=(Value of 5th observation+Value of 6th observation) /2

=(26+27)/2

=26.5

Sample Variance

Sample Variance S2=(∑dx^2-((∑dx)^2)/n ) /n-1

=(657-(1)^2/ 10) /9

=657-0.1/9

=656.9/9

=72.99

SD=Sqrt(Var)

SD=8.5434

t test:

Given values:

        

            (1) The provided sample means are :

                                            X̅1 = 44.5

                                            X̅2 = 28.1

                                       

                and the known population standard deviations are :

S1 = 8.683

S2 = 8.543

                and the sample size are n1 = 10 and n2 = 10.

            (2) Our test hypothesis is :

                The following null and alternative hypothesis need to be tested,

                hypothesis =

                h0: μ1 = μ2 or Older Adults = Younger Adults

H1: μ1 ≠ μ1 or  Older Adults != Younger Adults

                This hypothesis corresponds to a two-tailed, for which a t-test for two population means, with two

                independent samples, with unknown population standard deviations will be used.

            (3) Test statistics :

                The calculation of the t-test proceeds as follows,

                z = X̅1 - X̅2                    44.5 - 28.1

                   ---------------------- = -------------------------------- = 4.2576

                    √(σ1²/n1)+(σ2²/n2)      √(8.683²/10) + (8.543²/10)

            (4) Rejection Criteria :

         Based on the information provided, the significance level is α = 0.05,and the degrees of freedom are df= 18

                and the critical value for a two-tailed test is t tabulated = 1.7341

                And the rejection region for this two-tailed test is R =[t: |t| > 1.7341]

            (5) Decision about the null hypothesis :

                Since it is observed that t calculated means |t| = 4.2576 > t tabulated = 1.7341

                it is then concluded that the null hypothesis is Rejected.

                Using the P-value approach:

                The p-value is 0.0005, and since p = 0.0005 < α = 0.05

                it is then concluded that the null hypothesis is Rejected.

           

            (6) Conclusion

                It is concluded that the null hypothesis Ho is Rejected. Therefore, there is enough evidence to claim that

                population mean Older Adults is different than Younger Adults  at the 0.05 significance level.