Question

Group Statistics depression status of participants Mean Std. Deviation mean number of hours Depressed patient slept for 3 nights 8 6.9125 43895 Control 8 7.5500 52099 Group Statistics depression status of participants Std. Eror Mean mean number of hours Depressed patient slept for 3 nights 15519 Control .18420 Independent Samples Test Levene's Test for Equality of Variances t-test for Equality . Sig mean number of hours slept for 3 nights Equal variances assumec .008 .929 -2.647 Equal variances not assumed -2.647

2. Look at the Independent samples t-test table to get your t calculated, df, and significance. Read the line that says “equal variances assumed”. Remember that significance level will appear in the Sig (2-tailed) column. Any value that is smaller than .05 will be significant at the .05 level or higher. In a sentence describe the results and report them in APA style, including r² and evaluation (you will need to calculate r² by hand).

3. What type of error in hypothesis testing is possible given the above result? How likely is this error given r²?

Answer 1

2.

$Let~\mu_d=population~mean~no.~of~hours~slept~by~depressed~patient\\ \mu_c=population~mean~no.~of~hours~slept~by~control.\\ Null~hypothesis,~H_0:\mu_d=\mu_c~vs.~Alternative~hypothesis,~H_1:\mu_d\neq \mu_c\\ Assume~population~variances~are~equal~and~unknown~so~we~estimate~it~by~\\ sample~pooled~variance,~s^2=\frac{(8-1)s_1^2+(8-1)s_2^2}{8+8-2}\\ =7*(0.43895^2+0.52099^2)/14= 0.232\\ where,~s_1^2=0.43895^2,~s_2^2=0.52099^2\\ Test~statistic=t=\frac{\bar{x}_1-\bar{x}_2}{s\sqrt{\frac{1}{8}+\frac{1}{8}}}=-2.647,~where,~\bar{x}_1=6.9125,~\bar{x}_2=7.5500$

$p-value=2P(t>2.647|t\sim t_{14})=0.0191\\ ~where,~t\sim~t~distribution~with~14~df~under~H_0.\\ Since~p-value<0.05,~so~we~reject~H_0~at~level~0.05~and~conclude~that\\ there~is~significant~difference~between~mean~no.~of~hr.~slept~by~depressed\\ patients~and~control.\\ 3.~Since~we~reject~H_0~on~the~basis~of~sample~however~in~reality~this~difference\\ may~not~be ~significant~so~Type~I~error~is~committed.$