Recall that the sample mean $M$ and sample variance $S^2$ are $$M=\frac{1}{n}\sum _{i=1}^n{X}_i,S^2=\frac{1}{n-1}\sum _{i=1}^n(X_i-M{)}^2$$

Define $$Z=\frac{M-\mu }{\sigma /\sqrt{n}},T=\frac{M-\mu }{S/\sqrt{n}},V=\frac{n-1}{{\sigma }^2}{S}^2$$

1. $Z$ has the standard normal distribution.
2. $T$ has the student $t$ distribution with $n-1$ degrees of freedom.
3. $V$ has the chi-square distribution with $n-1$ degrees of freedom.
4. $Z$ and $V$ are independent.

Let $p\in (0,1)$ and $k\in {\mathbb{N}}_{+}$.

1. $z(p)$ denotes the quantile of order $p$ for the standard normal distribution.
2. $t_k(p)$ denotes the quantile of order $p$ for the student $t$ distribution with $k$ degrees of freedom.
3. ${\chi }_k^2(p)$ denotes the quantile of order $p$ for the chi-square distribution with $k$ degrees of freedom

For a conjectured ${\mu }_0\in \mathbb{R}$, define the test statistic $$Z=\frac{M-{\mu }_0}{\sigma /\sqrt{n}}$$

1. If $\mu ={\mu }_0$ then $Z$ has the standard normal distribution.
2. If $\mu \ne {\mu }_0$ then $Z$ has the normal distribution with mean $\frac{\mu -{\mu }_0}{\sigma /\sqrt{n}}$ and variance 1.

For $\alpha \in (0,1)$, each of the following tests has significance level $\alpha$:

1. Reject $H_0:\mu ={\mu }_0$ versus $H_1:\mu \ne {\mu }_0$ if and only if $Z<-z(1-\alpha /2)$ or $Z>z(1-\alpha /2)$ if and only if $M<{\mu }_0-z(1-\alpha /2)\frac{\sigma }{\sqrt{n}}$ or $M>{\mu }_0+z(1-\alpha /2)\frac{\sigma }{\sqrt{n}}$.
2. Reject $H_0:\mu \le {\mu }_0$ versus $H_1:\mu >{\mu }_0$ if and only if $Z>z(1-\alpha )$ if and only if $M>{\mu }_0+z(1-\alpha )\frac{\sigma }{\sqrt{n}}$.
3. Reject $H_0:\mu \ge {\mu }_0$ versus $H_1:\mu <{\mu }_0$ if and only if $Z<-z(1-\alpha )$ if and only if $M<{\mu }_0-z(1-\alpha )\frac{\sigma }{\sqrt{n}}$.

For each of the tests in [7], we fail to reject $H_0$ at significance level $\alpha$ if and only if ${\mu }_0$ is in the corresponding $1-\alpha$ confidence interval, that is

1. $M-z(1-\alpha /2)\frac{\sigma }{\sqrt{n}}\le {\mu }_0\le M+z(1-\alpha /2)\frac{\sigma }{\sqrt{n}}$
2. ${\mu }_0\le M+z(1-\alpha )\frac{\sigma }{\sqrt{n}}$
3. ${\mu }_0\ge M-z(1-\alpha )\frac{\sigma }{\sqrt{n}}$

For every $\alpha ,p\in (0,1)$, the following test has significance level $\alpha$: Reject $H_0:\mu ={\mu }_0$ versus $H_1:\mu \ne {\mu }_0$ if and only if $Z<z(\alpha -p\alpha )$ or $Z\ge z(1-p\alpha )$.

1. $p=\frac{1}{2}$ gives the symmetric, unbiased test.
2. $p\downarrow 0$ gives the left-tailed test.
3. $p\uparrow 1$ gives the right-tailed test.

The $P$-values of the standard tests in [20] are respectively

1. $2[1-\mathrm{\Phi }\left(\left|Z\right|\right)]$
2. $1-\mathrm{\Phi }(Z)$
3. $\mathrm{\Phi }(Z)$

The power function of the general two-sided test in [7] is given by $$Q(\mu )=\mathrm{\Phi }(z(\alpha -p\alpha )-\frac{\sqrt{n}}{\sigma }(\mu -{\mu }_0))+\mathrm{\Phi }(\frac{\sqrt{n}}{\sigma }(\mu -{\mu }_0)-z(1-p\alpha )),\mu \in \mathbb{R}$$

1. $Q$ is decreasing on $(-\mathrm{\infty },m_0)$ and increasing on $(m_0,\mathrm{\infty })$ where $m_0={\mu }_0+[z(\alpha -p\alpha )+z(1-p\alpha )]\frac{\sqrt{n}}{2\sigma }$.
2. $Q({\mu }_0)=\alpha$.
3. $Q(\mu )\to 1$ as $\mu \uparrow \mathrm{\infty }$ and $Q(\mu )\to 1$ as $\mu \downarrow -\mathrm{\infty }$.
4. If $p=\frac{1}{2}$ then $Q$ is symmetric about ${\mu }_0$ (and $m_0={\mu }_0$).
5. As $p$ increases, $Q(\mu )$ increases if $\mu >{\mu }_0$ and decreases if $\mu <{\mu }_0$.

The power function of the left-tailed test in [7] is given by $$Q(\mu )=\mathrm{\Phi }(z(\alpha )+\frac{\sqrt{n}}{\sigma }(\mu -{\mu }_0)),\mu \in \mathbb{R}$$

1. $Q$ is increasing on $\mathbb{R}$.
2. $Q({\mu }_0)=\alpha$.
3. $Q(\mu )\to 1$ as $\mu \uparrow \mathrm{\infty }$ and $Q(\mu )\to 0$ as $\mu \downarrow -\mathrm{\infty }$.

The power function of the right-tailed test in [7], is given by $$Q(\mu )=\mathrm{\Phi }(z(\alpha )-\frac{\sqrt{n}}{\sigma }(\mu -{\mu }_0)),\mu \in \mathbb{R}$$

1. $Q$ is decreasing on $\mathbb{R}$.
2. $Q({\mu }_0)=\alpha$.
3. $Q(\mu )\to 0$ as $\mu \uparrow \mathrm{\infty }$ and $Q(\mu )\to 1$ as $\mu \downarrow -\mathrm{\infty }$.

For any of the three tests in above in [7], increasing the sample size $n$ or decreasing the standard deviation $\sigma$ results in a uniformly more powerful test.

In the mean test experiment, select the normal test statistic and select the normal sampling distribution with standard deviation $\sigma =2$ sample size $n=20$, and ${\mu }_0=0$. Run the experiment 1000 times for several values of the true distribution mean $\mu$. For each value of $\mu$, note the distribution of the $P$-value.

In the mean estimate experiment, select the normal pivot variable and select the normal distribution with $\mu =0$ and standard deviation $\sigma =2$, confidence level $1-\alpha =0.90$, and sample size $n=10$. For each of the three types of confidence intervals, run the experiment 20 times. State the corresponding hypotheses and significance level, and for each run, give the set of ${\mu }_0$ for which the null hypothesis would be rejected.

For either of the one-sided tests in [7], the sample size $n$ needed for a test with significance level $\alpha$ and power $\beta$ for the alternative ${\mu }_1$ is $$n={\left(\frac{\sigma [z(\beta )-z(\alpha )]}{{\mu }_1-{\mu }_0}\right)}^2$$

For the unbiased, two-sided test, the sample size $n$ needed for a test with significance level $\alpha$ and power $\beta$ for the alternative ${\mu }_1$ is approximately $$n={\left(\frac{\sigma [z(\beta )-z(\alpha /2)]}{{\mu }_1-{\mu }_0}\right)}^2$$

For a conjectured ${\mu }_0\in \mathbb{R}$, define the test statistic $$T=\frac{M-{\mu }_0}{S/\sqrt{n}}$$

1. If $\mu ={\mu }_0$, the statistic $T$ has the student $t$ distribution with $n-1$ degrees of freedom.
2. If $\mu \ne {\mu }_0$ then $T$ has a non-central $t$ distribution with $n-1$ degrees of freedom and non-centrality parameter $\frac{\mu -{\mu }_0}{\sigma /\sqrt{n}}$.

For $\alpha \in (0,1)$, each of the following tests has significance level $\alpha$:

1. Reject $H_0:\mu ={\mu }_0$ versus $H_1:\mu \ne {\mu }_0$ if and only if $T<-t_{n-1}(1-\alpha /2)$ or $T>t_{n-1}(1-\alpha /2)$ if and only if $M<{\mu }_0-t_{n-1}(1-\alpha /2)\frac{S}{\sqrt{n}}$ or $T>{\mu }_0+t_{n-1}(1-\alpha /2)\frac{S}{\sqrt{n}}$.
2. Reject $H_0:\mu \le {\mu }_0$ versus $H_1:\mu >{\mu }_0$ if and only if $T>t_{n-1}(1-\alpha )$ if and only if $M>{\mu }_0+t_{n-1}(1-\alpha )\frac{S}{\sqrt{n}}$.
3. Reject $H_0:\mu \ge {\mu }_0$ versus $H_1:\mu <{\mu }_0$ if and only if $T<-t_{n-1}(1-\alpha )$ if and only if $M<{\mu }_0-t_{n-1}(1-\alpha )\frac{S}{\sqrt{n}}$.

For each of the tests in [18], we fail to reject $H_0$ at significance level $\alpha$ if and only if ${\mu }_0$ is in the corresponding $1-\alpha$ confidence interval.

1. $M-t_{n-1}(1-\alpha /2)\frac{S}{\sqrt{n}}\le {\mu }_0\le M+t_{n-1}(1-\alpha /2)\frac{S}{\sqrt{n}}$
2. ${\mu }_0\le M+t_{n-1}(1-\alpha )\frac{S}{\sqrt{n}}$
3. ${\mu }_0\ge M-t_{n-1}(1-\alpha )\frac{S}{\sqrt{n}}$

For every $\alpha ,p\in (0,1)$, the following test has significance level $\alpha$: Reject $H_0:\mu ={\mu }_0$ versus $H_1:\mu \ne {\mu }_0$ if and only if $T<t_{n-1}(\alpha -p\alpha )$ or $T\ge t_{n-1}(1-p\alpha )$ if and only if $M<{\mu }_0+t_{n-1}(\alpha -p\alpha )\frac{S}{\sqrt{n}}$ or $M>{\mu }_0+t_{n-1}(1-p\alpha )\frac{S}{\sqrt{n}}$.

1. $p=\frac{1}{2}$ gives the symmetric, unbiased test.
2. $p\downarrow 0$ gives the left-tailed test.
3. $p\uparrow 1$ gives the right-tailed test.

The $P$-values of the standard tests in [20] are respectively

1. $2[1-{\mathrm{\Phi }}_{n-1}\left(\left|T\right|\right)]$
2. $1-{\mathrm{\Phi }}_{n-1}(T)$
3. ${\mathrm{\Phi }}_{n-1}(T)$

In the mean test experiment, select the student test statistic and select the normal sampling distribution with standard deviation $\sigma =2$, sample size $n=20$, and ${\mu }_0=1$. Run the experiment 1000 times for several values of the true distribution mean $\mu$. For each value of $\mu$, note the empirical distribution of $\mathbb{P}$.

In the mean estimate experiment, select the student pivot variable and select the normal sampling distribution with mean 0 and standard deviation 2. Select confidence level 0.90 and sample size 10. For each of the three types of intervals, run the experiment 20 times. State the corresponding hypotheses and significance level, and for each run, give the set of ${\mu }_0$ for which the null hypothesis would be rejected.

For a conjectured value ${\sigma }_0\in (0,\mathrm{\infty })$, define the test statistic $$V=\frac{n-1}{{\sigma }_0^2}{S}^2$$

1. If $\sigma ={\sigma }_0$, then $V$ has the chi-square distribution with $n-1$ degrees of freedom.
2. If $\sigma \ne {\sigma }_0$ then $V$ has the gamma distribution with shape parameter $(n-1)/2$ and scale parameter $2{\sigma }^2/{\sigma }_0^2$.

For every $\alpha \in (0,1)$, the following test has significance level $\alpha$:

1. Reject $H_0:\sigma ={\sigma }_0$ versus $H_1:\sigma \ne {\sigma }_0$ if and only if $V<{\chi }_{n-1}^2(\alpha /2)$ or $V>{\chi }_{n-1}^2(1-\alpha /2)$ if and only if $S^2<{\chi }_{n-1}^2(\alpha /2)\frac{{\sigma }_0^2}{n-1}$ or $S^2>{\chi }_{n-1}^2(1-\alpha /2)\frac{{\sigma }_0^2}{n-1}$
2. Reject $H_0:\sigma \ge {\sigma }_0$ versus $H_1:\sigma <{\sigma }_0$ if and only if $V<{\chi }_{n-1}^2(\alpha )$ if and only if $S^2<{\chi }_{n-1}^2(\alpha )\frac{{\sigma }_0^2}{n-1}$
3. Reject $H_0:\sigma \le {\sigma }_0$ versus $H_1:\sigma >{\sigma }_0$ if and only if $V>{\chi }_{n-1}^2(1-\alpha )$ if and only if $S^2>{\chi }_{n-1}^2(1-\alpha )\frac{{\sigma }_0^2}{n-1}$

For each of the tests in [25], we fail to reject $H_0$ at significance level $\alpha$ if and only if ${\sigma }_0^2$ is in the corresponding $1-\alpha$ confidence interval. That is

1. $\frac{n-1}{{\chi }_{n-1}^2(1-\alpha /2)}{S}^2\le {\sigma }_0^2\le \frac{n-1}{{\chi }_{n-1}^2(\alpha /2)}{S}^2$
2. ${\sigma }_0^2\le \frac{n-1}{{\chi }_{n-1}^2(\alpha )}{S}^2$
3. ${\sigma }_0^2\ge \frac{n-1}{{\chi }_{n-1}^2(1-\alpha )}{S}^2$

For every $\alpha ,p\in (0,1)$, the following test has significance level $\alpha$: Reject $H_0:\sigma ={\sigma }_0$ versus $H_1:\sigma \ne {\sigma }_0$ if and only if $V\le {\chi }_{n-1}^2(\alpha -p\alpha )$ or $V\ge {\chi }_{n-1}^2(1-p\alpha )$ if and only if $S^2<{\chi }_{n-1}^2(\alpha -p\alpha )\frac{{\sigma }_0^2}{n-1}$ or $S^2>{\chi }_{n-1}^2(1-p\alpha )\frac{{\sigma }_0^2}{n-1}$.

1. $p=\frac{1}{2}$ gives the equal-tail test.
2. $p\downarrow 0$ gives the left-tail test.
3. $p\uparrow 1$ gives the right-tail test.

The power function of the general two-sided test in [27] is given by the following formula, and satisfies the given properties: $$Q(\sigma )=1-G_{n-1}(\frac{{\sigma }_0^2}{{\sigma }^2}{\chi }_{n-1}^2(1-p\alpha ))+G_{n-1}(\frac{{\sigma }_0^2}{{\sigma }^2}{\chi }_{n-1}^2(\alpha -p\alpha ))$$

1. $Q$ is decreasing on $(-\mathrm{\infty },{\sigma }_0)$ and increasing on $({\sigma }_0,\mathrm{\infty })$.
2. $Q({\sigma }_0)=\alpha$.
3. $Q(\sigma )\to 1$ as $\sigma \uparrow \mathrm{\infty }$ and $Q(\sigma )\to 1$ as $\sigma \downarrow 0$.

The power function of the left-tailed test in [25] is given by the following formula, and satisfies the given properties: $$Q(\sigma )=1-G_{n-1}(\frac{{\sigma }_0^2}{{\sigma }^2}{\chi }_{n-1}^2(1-\alpha ))$$

1. $Q$ is increasing on $(0,\mathrm{\infty })$.
2. $Q({\sigma }_0)=\alpha$.
3. $Q(\sigma )\to 1$ as $\sigma \uparrow \mathrm{\infty }$ and $Q(\sigma )\to 0$ as $\sigma \downarrow 0$.

The power function for the right-tailed test in [25] is given by the following formula, and satisfies the given properties: $$Q(\sigma )=G_{n-1}(\frac{{\sigma }_0^2}{{\sigma }^2}{\chi }_{n-1}^2(\alpha ))$$

1. $Q$ is decreasing on $(0,\mathrm{\infty })$.
2. $Q({\sigma }_0)=\alpha$.
3. $Q(\sigma )\to 0$ as $\sigma \uparrow \mathrm{\infty })$ and $Q(\sigma )\to 0$ as $\sigma \uparrow \mathrm{\infty }$ and as $\sigma \downarrow 0$.

In the variance test experiment, select the normal distribution with mean 0, and select significance level 0.1, sample size 10, and test standard deviation 1.0. For various values of the true standard deviation, run the simulation 1000 times. Record the relative frequency of rejecting the null hypothesis and plot the empirical power curve.

1. Two-sided test
2. Left-tailed test
3. Right-tailed test

In the variance estimate experiment, select the normal distribution with mean 0 and standard deviation 2, and select confidence level 0.90 and sample size 10. Run the experiment 20 times. State the corresponding hypotheses and significance level, and for each run, give the set of test standard deviations for which the null hypothesis would be rejected.

1. Two-sided confidence interval
2. Confidence lower bound
3. Confidence upper bound

In the mean test experiment, select the gamma distribution with shape parameter 1 and scale parameter 1. For the three different tests and for various sample sizes, and values of ${\mu }_0$, run the experiment 1000 times. For each configuration, note the empirical distribution of $\mathbb{P}$.

In the mean test experiment, select the uniform distribution on $[0,4]$. For the three different tests and for various sample sizes and values of ${\mu }_0$, run the experiment 1000 times. For each configuration, note the empirical distribution of $\mathbb{P}$.

In the variance test experiment, select the gamma distribution with shape parameter 1 and scale parameter 1. For the three different tests and for various significance levels, sample sizes, and values of ${\sigma }_0$, run the experiment 1000 times. For each configuration, note the relative frequency of rejecting $H_0$. When $H_0$ is true, compare the relative frequency with the significance level.

In the variance test experiment, select the uniform distribution on $[0,4]$. For the three different tests and for various significance levels, sample sizes, and values of ${\mu }_0$, run the experiment 1000 times. For each configuration, note the relative frequency of rejecting $H_0$. When $H_0$ is true, compare the relative frequency with the significance level.

The length of a certain machined part is supposed to be 10 centimeters. In fact, due to imperfections in the manufacturing process, the actual length is a random variable. The standard deviation is due to inherent factors in the process, which remain fairly stable over time. From historical data, the standard deviation is known with a high degree of accuracy to be 0.3. The mean, on the other hand, may be set by adjusting various parameters in the process and hence may change to an unknown value fairly frequently. We are interested in testing $H_0:\mu =10$ versus $H_1:\mu \ne 10$.

1. Suppose that a sample of 100 parts has mean 10.1. Perform the test at the 0.1 level of significance.
2. Compute the $P$-value for the data in (a).
3. Compute the power of the test in (a) at $\mu =10.05$.
4. Compute the approximate sample size needed for significance level 0.1 and power 0.8 when $\mu =10.05$.

A bag of potato chips of a certain brand has an advertised weight of 250 grams. Actually, the weight (in grams) is a random variable. Suppose that a sample of 75 bags has mean 248 and standard deviation 5. At the 0.05 significance level, perform the following tests:

1. $H_0:\mu \ge 250$ versus $H_1:\mu <250$
2. $H_0:\sigma \ge 7$ versus $H_1:\sigma <7$

At a telemarketing firm, the length of a telephone solicitation (in seconds) is a random variable. A sample of 50 calls has mean 310 and standard deviation 25. At the 0.1 level of significance, can we conclude that

1. $\mu >300$?
2. $\sigma >20$?

At a certain farm the weight of a peach (in ounces) at harvest time is a random variable. A sample of 100 peaches has mean 8.2 and standard deviation 1.0. At the 0.01 level of significance, can we conclude that

1. $\mu >8$?
2. $\sigma <1.5$?

The hourly wage for a certain type of construction work is a random variable with standard deviation 1.25. For sample of 25 workers, the mean wage was $6.75. At the 0.01 level of significance, can we conclude that $\mu <7.00$?

Using Michelson's data, test to see if the velocity of light is greater than 730 (+299000) km/sec, at the 0.005 significance level.

Using Cavendish's data, test to see if the density of the earth is less than 5.5 times the density of water, at the 0.05 significance level .

Using Short's data, test to see if the parallax of the sun differs from 9 seconds of a degree, at the 0.1 significance level.

Using Fisher's iris data, perform the following tests, at the 0.1 level:

1. The mean petal length of Setosa irises differs from 15 mm.
2. The mean petal length of Verginica irises is greater than 52 mm.
3. The mean petal length of Versicolor irises is less than 44 mm.