Question

The data below comes from a university that wanted to see if there was a relationship between IQ scores and the GPA's of students after their freshman year. Use the data for questions 1 through 5 For each hypothesis test, provide the daim, critical values, test statistic and condusion 1) using a 95% level of confidence, test the dam that IQ's have a mean of 100. 2) Using a 95% level of confidence, test the daim that the majority of students have at least a 3.0 GPA. 3) At the 98% confidence level, test the claim that mean GPA for freshmen is over 3.0. 4) Test the claim that the standard deviation for IQ scores is 15. 5) Divide the data into two groups: those with under a 110IQ and those with at least 110 1Q. a) Test the claim that the mean GPAis 3.0 for those with under a 110 IQ b) Test the claim that the mean GPA is 3.0 for those with at least 110 1Q.

Answer 1

$1.\\ Null~hypothesis,~H_0:\mu_1=100~vs.~Alternative~hypothesis,~H_1:\mu_1\neq 100.\\ where~\mu_1=population~mean~of~IQ's\\ Critical~values=\pm t_{0.025,24}=\pm 2.06\\ \bar{x}_1=sample~mean~of~IQ's=114.24,~s_1=sample~sd~of~IQ's=14.10~then\\ test~statistic=\frac{\sqrt{25}(\bar{x}_1-100)}{s_1}=5.05\\ Since~value~of~test~statistic>2.06~so~we~reject~H_0~with~95\%~confident~and\\ conclude~that~there~is~insufficient~evidence~to~accept~the~claim~that~IQ's\\ have~a~mean~of~100.\\ 2.\\ Null~hypothesis,~H_0:\mu_2=3.0~vs.~Alternative~hypothesis,~H_1:\mu_2<3.0.\\ where~\mu_2=population~mean~of~GPA's\\ Critical~value= -t_{0.05,24}=-1.71\\ \bar{x}_2=sample~mean~of~GPA's= 3.3032,~s_2=sample~sd~of~IQ's=0.4934~then\\ test~statistic=\frac{\sqrt{25}(\bar{x}_2-3)}{s_2}= 3.07\\ Since~value~of~test~statistic>-1.71~so~we~accept~H_0~with~95\%~confident~and\\ conclude~that~there~is~sufficient~evidence~to~accept~the~claim~that~the~majority\\ of~students~have~at~least~a~3.0~GPA.\\$

$3.\\ Null~hypothesis,~H_0:\mu_2=3.0~vs.~Alternative~hypothesis,~H_1:\mu_2>3.0.\\ Critical~value=t_{0.02,24}=2.17\\ test~statistic=\frac{\sqrt{25}(\bar{x}_2-3)}{s_2}= 3.07\\ Since~value~of~test~statistic>2.17~so~we~reject~H_0~with~98\%~confident~and\\ conclude~that~there~is~sufficient~evidence~to~accept~the~claim~that~mean~GPA\\ for~freshmen~is~over~3.0~GPA.\\$

$4.\\ Null~hypothesis,~H_0:\sigma_1=15~vs.~Alternative~hypothesis,~H_1:\sigma_1\neq 15.\\ where~\sigma_1=population~sd~of~IQ's\\ Critical~values=\chi^2_{0.025,24}=39.36~and~\chi^2_{0.975,24}=12.40\\ test~statistic=\chi^2=\frac{24s_1^2}{15^2}=21.19\\ Since~12.40<value~of~test~statistic<39.36~so~we~fail~to~reject~H_0~with~95\%~confident~and\\ conclude~that~there~is~sufficient~evidence~to~accept~the~claim~that~the\\ standard~deviation~of~IQ~scores~is~15.\\ 5.~(a)\\Null~hypothesis,~H_0:\mu_1^*=3~vs.~Alternative~hypothesis,~H_1:\mu_1^*\neq 3.\\ Critical~values=\pm t_{0.025,11}=\pm 2.20\\ \mu_1^*=true~mean~GPA~of~students~with~under~a~110~IQ\\ \bar{y}_1=sample~mean~GPA~for~those~with~under~a~110~IQ= 2.8750\\ s_1^*=sample~sd~of~GPAs~for~those~with~under~a~110~IQ= 0.3176\\ Test~statistic=\frac{\sqrt{12}(\bar{y}_1-3)}{s_1^*}= -1.36.\\ Since~-2.20<Value~of~test~statistic<2.20~so\\ so~we~fail~to~reject~H_0~with~95\%~confident~and\\ conclude~that~there~is~sufficient~evidence~to~accept~the~claim.\\$

$(b)\\Null~hypothesis,~H_0:\mu_2^*=3~vs.~Alternative~hypothesis,~H_1:\mu_2^*\neq 3.\\ \mu_2^*=true~mean~GPAs~of~the~students~with~at`least~110~IQ.\\ Critical~values=\pm t_{0.025,12}=\pm 2.18\\ \bar{y}_2=sample~mean~GPA~for~those~with~at least~110~IQ= 3.6985\\ s_2^*=sample~sd~of~GPAs~for~those~with~at~least~110~IQ= 0.2044\\ Test~statistic=\frac{\sqrt{13}(\bar{y}_2-3)}{s_2^*}= 12.32.\\ Since~-2.20<Value~of~test~statistic<2.20~so\\ so~we~reject~H_0~with~95\%~confident~and\\ conclude~that~there~is~sufficient~evidence~to~reject~the~claim.\\$