---
tags: 機率與統計
title: 第十四週活動
---

# 活動一
1. State whether each of the following assertions is a legitimate statistical hypothesis and why:

(a) H: σ>100
O Yes

(b) H: x¯ =45
X No

(c) H: $\frac{σ1}{σ2}$ < 1
O Yes

(d) H: x1¯ -x2¯= -5.0
X No

(e) H: x = sound intensity of a certain source (decibels) has a normal distribution.
O Yes

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# 活動二
A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is for proposed that the new design be implemented only if simple data strongly indicates a reduction in average braking distance for the new design. State relevant hypotheses and describes the type I and type II errors in the context of this situation.

$H_0 = 120$
$H_1 < 120$

Type 1 error
>相信$H_1$成立，可以推出新系統優於就系統，但實際上$H_0$才是對的

Type 2 error
>不相信$H_1$成立，可以推出舊系統可能優於新系統，但實際上$H_1$才是對的

# 活動三

Newly purchased automobile tires of a certain type are supposed to be filled to a pressure of 34 psi. Let μ note the true average pressure. A test of H_0: μ = 34 versus H_a μ ≠ 34 will be based on a large simple of tires so that the test statistic x = (x¯−34)/(sn‾√) will have approximately a standard normal distribution when H_0 is true. Determine the value of z and that the P-value in each of the following cases:
>$H_0=34,H_1\neq 34$
>$\frac{\overline{x}-u}{(\frac{s}{\sqrt{n}})}=z$
>$\alpha=0.05$
>$0.975=>z=1.96$

(a) $n=50, \overline x=34.43,s=1.06$
>$z=\frac{\overline{x}-34}{\frac{1.06}{\sqrt 50}}=2.86$
>$2.86>1.96$
>拒絕 $H_0$
>$z = 2.86,P_{value} = 1-0.9979 = 0.0021$

(b) $n=36, \overline{x}=34.66,s=2.53$
>$Z=\frac{\overline{x}-34}{\frac{2.53}{\sqrt 36}}=1.56$
>$Z=1.56,P_{value} = 0.0594$
>不拒絕$H_0$