Probability and Statistics Practical Class, Week 19

Question One §

Question §

Given the following data following array of numbers…

$$5.9,6.3,6.3,6.5,6.8,6.8,7.0,7.0,7.2,7.3,7.4,7.6,7.7,7.7,7.8,7.8,7.9,8.1,8.2,8.7,9.0,9.7,9.7,10.7,11.3,11.6,11.8$$

First, construct a stem-and-leaf display. Then answer the following equations:

1. What appears to be a representative value? Do the observations appear to be highly concentrated about the representative value or rather spread out?

2. Does the display appear to be reasonably symmetric about a representative value, or would you describe its shape in some other way?

3. Do there appear to be any outlying values?

4. What proportion of observations in this sample exceed 10?

Solution §

stem-and-leaf display

1. Around about $7.8$ seems fairly representative, although there is a fair bit of variation around the value.

2. The display is reasonably symmetric around $7.8$, however there’s another increase in concentration around 11 - still, there is only one significant peak (at the representative value).

3. There appears to be no outliers.

4. $\frac{4}{27}\approx 14.8\%$ of the observations exceed 10.

Question Two §

Question §

How does the speed of a runner vary over the course of a marathon (a distance of $42.195\text{km}$)? Consider determining both th time to run the first $5$km and the time to run between the $35\text{km}$ and $40\text{km}$ points, and then subtracting the former time from the latter time. A positive value of this difference corresponds to a runner slowing down toward the end of the race. The accompanying histogram is based on times of runners who participated in several different Japanese marathons (“Factors Affecting Runners’ Marathon Performance,” Chance, Fall, 1993: 24-30).

Given this, answer the following questions:

1. What are some interesting features of this histogram?

2. What is a typical difference value?

3. Roughly what proportion of the runners ran the late distance more quickly than the early distance?

Solution §

1. Clear symmetric distribution around the time difference of $100$ and the other smaller numbers, rather than the tail-end where there are minimal runners with a time difference greater than $400$: it has a very large positive skew (a lot of runners slowed down in the last $5\text{km}$.

2. A typical difference value is between $50$ and $100$.

3. A very small proportion: around $1\%$.

Question Three §

Question §

The article “Statistical Modelling of the Time Course of Tantrum Anger?” (Annals of Applied Stats, 2009: 1014-1034) discussed how anger intensity in children’s tantrums could be related to tantrum duration as well as behavioural indicators such as shouting, stamping, and pushing or pulling. The following frequency distribution was given:

Range	Frequency
$0\le x<2$	136
$2\le x<4$	92
$4\le x<11$	71
$11\le x<20$	26
$20\le x<30$	7
$30\le x<40$	3
Based on this, construct a corresponding histogram.

Solution §

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Question Four §

Question §

Give a possible sample space $\mathbb{S}$ for each of the following experiments:

1. An election decides between two candidates A and B.

2. A two-sided coin is tossed.

3. A student is asked for the month of the year and the day of the week on which her birthday falls.

4. A student is chosen at random from a class of ten students.

Solution §

1. ${\mathbb{S}}_1=\{A,B\}$.

2. ${\mathbb{S}}_2=\{H,T\}$.

3. ${\mathbb{S}}_3=\{\text{months}\times \text{days}\}$.

4. ${\mathbb{S}}_4=\{1,2,3,4,5,6,7,8,9,10\}$.

Question Five §

Question §

Let $\mathbb{S}=\{a,b,c\}$ be a sample space. Let $P(\{a\})=\frac{1}{2}$, $P(\{b\})=\frac{1}{3}$, $P(\{c\})=\frac{1}{6}$. Find the probabilities for all eight subsets of $\mathbb{S}$, such that the empty set $\varnothing$ is considered to be a subset of all sets.

Solution §

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Question Six §

Question §

A die is loaded in such a way that the probability of each face turning up is proportional to the number of dots on that face. For example, a six is three times as probable as a two.

1. Write down a sample space for the experiment of rolling this die.

2. Work out the probability function for this die.

3. What is the probability of getting an even number in one throw?

Solution §

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