Let X i be Gaussian random variables with μ = 10 and σ 2 = 1. We decide to use the test statistic μ ^ = 1 20 ∑ i = 1 20 X i and following tests: | μ ^ − 10 | &gt; τ: Reject H 0 | μ ^ − 10 | ≤ τ: Cannot reject H 0 1) Find τ if you want to have 95 % confidence in the test.2) You find that μ ^ = 10.588. Do you reject H 0 ? If so, what is the p-value?Using textbook equations, I found out that for1) τ = σ μ ^ = 1.96 1 20 = 0.3922) | 10.588 − 10 | = 0.588 &gt; 0.392 → Reject H 0 p-value = 0.003

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Let       X      i   be Gaussian random variables with   μ =   10 and       σ      2   =   1. We decide to use the test statistic       μ    ^   =       1    20        ∑          i      =      1          20        X      i   and following tests:  |      μ    ^    −  10  |  &amp;gt;  τ: Reject       H      0    |      μ    ^    −  10  |  ≤  τ: Cannot reject       H      0  1) Find   τ if you want to have   95  % confidence in the test.2) You find that       μ    ^    =  10.588. Do you reject       H      0  ? If so, what is the p-value?Using textbook equations, I found out that for1)   τ =       σ              μ      ^       =   1.96            1      20         =  0.3922)       |    10.588  −  10      |    =  0.588  &amp;gt;  0.392  → Reject       H    0  p-value   =  0.003

Kharroubip9ej0 · Accepted Answer

The   95  % confidence interval for   μ is:      μ    ^    −      z          α          /      2        ⋅      σ              μ      ^        &amp;lt;  μ  &amp;lt;      μ    ^    +      z          α          /      2        ⋅      σ              μ      ^          ⟺          μ    ^    −  1.96  ⋅      σ          n        &amp;lt;  μ  &amp;lt;      μ    ^    +  1.96  ⋅      σ          n          ⟺      −  1.96  ⋅      σ          n        &amp;lt;  μ  −      μ    ^    &amp;lt;  1.96  ⋅      σ          n          ⟺      |  μ  −      μ    ^    |  &amp;lt;  1.96  ⋅      σ          n        =  τ  =  1.96  ⋅      1          20        =  0.392     or        μ    ^    −  0.392  &amp;lt;  μ  &amp;lt;      μ    ^    +  0.392It implies that we are   95  % confident that the population mean will lie within   0.392 distance from       μ    ^  . The null hypothesis states to reject if it is greater distance. With       μ    ^    =  10.588, it is greater distance, so we reject       H    0  .  p-value is:  P      (    z    &amp;gt;                                  μ          ^                −                  μ          0                            σ              /                  n                      )    =  P      (    z    &amp;gt;                  10.588        −        10                    1              /                  20                      )    =  P  (  z  &amp;gt;  2.62962  )  =  0.004  &amp;lt;  0.025  ⇒         Reject             H      0        .

Let X i </msub> be Gaussian random variables with μ =

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