Standard +0.3 This is a straightforward one-sample hypothesis test with known variance. Students must calculate the sample mean (36.68), perform a two-tailed z-test at 1% significance level, and reach a conclusion. While it requires multiple steps (calculating mean, variance, test statistic, comparing to critical value), each step follows a standard algorithm with no conceptual challenges or novel problem-solving required. It's slightly easier than average because the setup is clear and the procedure is routine.
1.
The random variable \(X\) denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that \(\mathrm { E } ( X ) = 38.4\). Under a new treatment, the yields of 50 randomly chosen regions can be summarised as
$$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$
Test at the \(1 \%\) level whether there has been a change in the mean crop yield. [0pt]
1.
The random variable $X$ denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that $\mathrm { E } ( X ) = 38.4$. Under a new treatment, the yields of 50 randomly chosen regions can be summarised as
$$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$
Test at the $1 \%$ level whether there has been a change in the mean crop yield.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM Statistics 2021 Q1 [9]}}