Answer
a. $H_{o}: σ = 112$
$H_{1}: σ < 112$
b. Type-I Error: Sample evidence leads the teacher to believe that the standard deviation of SAT math scores has decreased. However, in fact it has not decreased.
c. Type-II Error: Sample evidence leads the teacher to believe that the standard deviation of SAT math scores has not decreased. However, in fact it did decrease.
Work Step by Step
a. Null hypothesis (the standard deviation of the test score is equal to 112):
Alternative hypothesis (the standard deviation of the test score is less than 112):
$H_{o}: σ = 112$
$H_{1}: σ < 112$
b. Type-I Error: Sample evidence leads the teacher to believe that the standard deviation of SAT math scores has decreased. However, in fact it has not decreased. In other words, the teacher has rejected the null hypothesis ($H_o$), but it in fact is true.
c. Type-II Error: Sample evidence leads the teacher to believe that the standard deviation of SAT math scores has not decreased. However, in fact it did decrease. In other words, the teacher has not rejected the null hypothesis ($H_o$), when in fact the alternative hypothesis ($H_1$) is true.
