In writing about experiments on the scattering of $\alpha$ particles in helium Rutherford said, "On account of the impossibility of distinguishing between the scattered alpha particles and the projected He nuclei, the results are subject to a certain ambiguity." Explain how an awareness of quantum statistics could have removed the ambiguity. What determines whether a gas obeys Bose or Fermi distributions?

Let $Y_1 < Y_2$ denote the order statistics of a random sample of size 2 from $N \left( 0 , \sigma ^ { 2 } \right)$ (a) Show that $E \left( Y _ { 1 } \right) = - \sigma / \sqrt { \pi }$ Hint: Evaluate $E(Y_1)$ by using the joint pdf of $Y_1$ and $Y_2$ and first integrating on $y_1.$ (b) Find the covariance of $Y_1$ and $Y_2.$

Here are the annual numbers of deaths from tornadoes in the United States from 1995 through 2015:

30, 25, 67, 130, 94, 41, 40, 55, 54, 35, 38, 67, 81, 126, 21, 45, 553, 70, 55, 47, 36

Find these statistics:

a) mean

b) median and quartiles

c) range and IQR

Use the following data set. The data represent the heights (in inches) of students in a statistics class.

$$\begin{array}{llllllllll}52 & 54 & 55 & 56 & 56 & 56 & 58 & 59 & 60 & 61 \\ 61 & 63 & 65 & 67 & 68 & 68 & 70 & 71 & 72 & \end{array}$$

Find the height that corresponds to the first quartile.

Here are the annual numbers of deaths from floods in the United States from 1995 through 2015:

80, 131, 118, 136, 68, 38, 48, 49, 86, 82, 43, 76, 87, 82, 56, 103, 113, 29, 82, 38, 176

Find these statistics:

a) mean

b) median and quartiles

c) range and IQR

Chemical analyses of 1599 red and 4898 white Italian wines showed the next summary statistics for $\mathrm{pH}$ (a measure of acidity):

Is there a difference in $\mathrm{pH}$ between red and white wines?

d) Is there a significant difference in $\mathrm{pH}$ ?

The Hopkins Forest dataset includes all 24 weather variables reported by the researchers. Many of the variables (e.g., temperature, relative humidity, solar radiation, wind) are reported as daily averages, minima and maxima. Using any of these variables, compare the distributions of the daily minima and maxima in a few sentences. Use summary statistics and appropriate graphical displays.

How do sleep problems affect teenagers? Choose one sleep disorder covered (such as insomnia, nightmares, sleepwalking, apnea, or narcolepsy) and use the Internet to research how the disorder affects teenagers. Look for information on how common the disorder is among teens, recent statistics and data, and common treatments. Write a short report summarizing your findings.

According to a study by Nielsen Mobile, "Teenagers ages 13 to 17 are by far the most prolific texters, sending 1742 messages a month" Mr. Williams, a high school statistics teacher, was skeptical about the claims in the article. So he collected data from his first-period statistics class on the number of text messages they had sent over the past 24 hours. Here are the data:

$$\begin{array}{rrrrrrrrrrrr} \hline 0 & 7 & 1 & 29 & 25 & 8 & 5 & 1 & 25 & 98 & 9 & 0 & 26 \\ 8 & 118 & 72 & 0 & 92 & 52 & 14 & 3 & 3 & 44 & 5 & 42 & \\ \hline \end{array}$$

Find the number of text messages sent by Joelle, who is at the 16th percentile of the distribution.

Identify the population, sample, population parameters, and sample statistics. In a test of the effectiveness of garlic in lowering cholesterol, $47$ adult subjects were treated with garlic in a processed tablet form. Cholesterol levels were measured before and after the treatment. The average (mean) change in the subjects' LDL cholesterol level was $3.2 \mathrm{mg} / \mathrm{dL}$ )

Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither:

Claim about IQ scores of statistics instructors: $\mu>100$. Sample data: $n=15$, $\bar{x}=118, s=11$. The sample data appear to come from a normally distributed population with unknown $\mu$ and $\sigma$.

Chemical analyses of 1599 red and 4898 white Italian wines showed the next summary statistics for $\mathrm{pH}$ (a measure of acidity):

Is there a difference in $\mathrm{pH}$ between red and white wines?

a) Write the null and alternative hypotheses

Determine whether a normal sampling distribution can be used. If can be used, test the claim about the difference between two population proportions

$$p _ { 1 }$$

and

$$p _ { 2 }$$

at the level of significance

$$\alpha$$

. Assume the samples are random and independent. Claim:

$$p _ { 1 } = p _ { 2 }; \alpha = 0.10$$

. Sample statistics:

$$x _ { 1 } = 42, n _ { 1 } = 150$$

and

$$x _ { 2 } = 76, n _ { 2 } = 200$$

According to the National Center for Health Statistics, the distribution of heights for 15-year-old males is modeled well by a Normal density curve. For this distribution, a z-score of 0 corresponds to a height of 170 centimeters ( cm}, and a z-score of 1 corresponds to a height of 177.5 cm. The distribution of heights for 16-year-old females is also modeled well by a Normal density curve. For this distribution, a z-score of 0 corresponds to a height of 162.5 cm, and a z-score of 1 corresponds to a height of 169 cm. Paul is 15 years old and 175 cm tall. (a) Find the z-score corresponding to Paul's height. Explain what this value means. (b) Paul's height puts him at the 75th percentile among 15-year-old males. Explain what this means to someone who knows no statistics.