MATH 2411 Homework 4

Q1

$$\begin{array}{rl}& H_0:\mu =10\\ & H_1:\mu <10\end{array}$$

Q2

a

Let the r.v. $X$ denotes the number of heads in 10 tosses, i.e. $X\sim \text{Binom}(10,p)$

$$\begin{array}{r}{H}_0:p=0.5\\ H_1:p\ne 0.5\end{array}$$

b

$$\begin{array}{rl}\text{p-value}& =P(X\ge 8)=1-P(X<8)=1-P(X\le 7)\\ & =1-\text{pbinom(7, 10, 0.5)}\\ & =0.0546875\end{array}$$

Q3

a

$$\begin{array}{rl}& H_0:\mu =100\\ & H_1:\mu \ge 100\end{array}$$

I agree with Bob’s analysis

Since $0.08256>0.05$, i.e. $\text{p-vaue}>\alpha$, we do not reject $H_0$, which means there is not enough evidence to reject the company’s claim

b

$$\begin{array}{rl}\text{t-value}& =\frac{\bar{x}-{\mu }_0}{s_{n-1}/\sqrt{n}}\\ 1.9098& =\frac{103.25-100}{s_{n-1}/\sqrt{12}}\\ s_{n-1}& =\frac{103.25-100}{1.9098}\times \sqrt{12}\\ & =5.895031\end{array}$$

c

$$t_{11,\frac{0.1}{2}}=\text{qt(0.05, df=11, lower.tail=F)}=1.795885$$ $$\begin{array}{rl}& \left[\bar{x}-t_{n-1,\alpha /2}\frac{s_{n-1}}{\sqrt{n}},\bar{x}+t_{n-1,\alpha /2}\frac{s_{n-1}}{\sqrt{n}}\right]\\ & =\left[103.25-1.795885\times \frac{5.895031}{\sqrt{12}},103.25+1.795885\times \frac{5.895031}{\sqrt{12}}\right]\\ & =\left[100.1939,106.3061\right]\end{array}$$

Q4

a

$$\begin{array}{rl}& H_0:\mu =6.5\\ & H_1:\mu \le 6.5\end{array}$$

x = read_excel("cold.xlsx")
t.test(x, alternative="less", mu=6.5, conf.level=0.95)

	One Sample t-test

data:  x
t = -0.20751, df = 9, p-value = 0.4201
alternative hypothesis: true mean is less than 6.5
95 percent confidence interval:
     -Inf 6.891683
sample estimates:
mean of x 
     6.45 

Since 0.4201 > 0.05, i.e. p-value > $\alpha$, we do not reject $H_0$

b

$$\begin{array}{rl}P\left(\frac{\bar{X}-{\mu }_1}{\sigma /\sqrt{n}}>\frac{{\mu }_0-z_{\alpha }\frac{\sigma }{\sqrt{n}}-{\mu }_1}{\sigma /\sqrt{n}}\right)& =\beta \\ P\left(Z>-z_{0.05}-\frac{7-6}{1/\sqrt{n}}\right)& =1-0.8\\ -z_{0.05}-\sqrt{n}& =0.2\\ -\text{qnorm(1-0.05)}-\sqrt{n}& =\text{qnorm(1-0.2)}\\ n& =6.182557\end{array}$$

Therefore we need 7 samples