On an elementary school examination in spelling, the mean grade of 32 boys was 72 with a standard deviation of 8, while the mean grade of 36 girls was 75 with a standard deviation of 6. Test the hypothesis at a (a) 0.05, (b) 0.01 level of significance that the girls are better in spelling than the boys

Answer:For Part A:Z>Z_{0.05} hence reject H₀ it means girls are better.Hₐ hypothesis is correctFor Part B:Z<Z_{0.01} hence Girls are not better and Hₙ hypothesis is correct.Step-by-step explanation:Consider the Two Hypothesis:H₀:[tex]u_{G}[/tex]=[tex]u_{B}[/tex]Hₐ:[tex]u_{G}[/tex]>[tex]u_{B}[/tex]Hₙ:[tex]u_{G}[/tex]<[tex]u_{B}[/tex]Test Statistics:Z=[tex]\frac{(x_{G} -x_{B} )-(u_{G}-u_{B)}  }{\sqrt{\frac{S_{G} ^{2} }{n_{G} }+\frac{S_{B}^{2} }{n_{B} }  } }[/tex]Where[tex]x_{G}[/tex] is the mean grade of girls[tex]x_{B}[/tex] is the mean grade of boys[tex]S_{G}[/tex] is the standard deviation of girls[tex]S_{B}[/tex] is the standard deviation of boys[tex]n_{G}[/tex] is the number of girls[tex]n_{B}[/tex] is the number of boysZ=[tex]\frac{(75 -72 )-0  }{\sqrt{\frac{6^{2} }{36 }+\frac{8^{2} }{32 }  } }[/tex]Z≅1.73Critical Value at [tex]Z_{0.05} = 1.645\\Z_{0.01} = 2.326[/tex]For Part A:Z>Z_{0.05} hence reject H₀ it means girls are better.Hₐ hypothesis is correctFor Part B:Z<Z_{0.01} hence Girls are not better and Hₙ hypothesis is correct.