1. The distribution of Scrabble tiles and their points are given below. As you might expect, there are more vowels and common consonants (e.g. R, T). High point tiles such as Q and Z occur less frequently. One game has 100 tiles total. Letter Point Value Frequency A 1 9 B 3 2 C 3 2 D 2 4 E 1 12 F 4 2 G 2 3 H 4 2 I 1 9 J 8 1 K 5 1 L 1 4 M 3 2 N 1 6 O 1 8 P 3 2 Q 10 1 R 1 6 S 1 4 T 1 6 U 1 4 V 4 2 W 4 2 X 8 1 Y 4 2 Z 10 1 Blank 0 2 Suppose we were to perform an experiment that consisted of drawing one tile from the bag. Let's define the random variable X as the number of points a tile is worth. a. Explain why X would be considered a discrete random variable b. What are the possible outcomes for the random variable X? c. What is the probability that a randomly selected tile is worth 0 points? That is , find P(X=0). d. What is the probability that a randomly selected tile is worth 4 points? That is , find P(X=4). e. Similar to (c) and (d), find the proabability for all possible outcomes and list them in the table below. Outcomes for X P(X=x) This table is known as the probability distribution of X. f. Using the table in (e), compute the probability that a tile is worth at most 3 points g. compute the probability that a tile is worth at least 2 points. h. find the expected (mean) number of points a Scrabble tile is worth.
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The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
1. The distribution of Scrabble tiles and their points are given below. As you might expect, there are more vowels and common consonants (e.g. R, T). High point tiles such as Q and Z occur less frequently. One game has 100 tiles total.
| Letter | Point Value | Frequency |
| A | 1 | 9 |
| B | 3 | 2 |
| C | 3 | 2 |
| D | 2 | 4 |
| E | 1 | 12 |
| F | 4 | 2 |
| G | 2 | 3 |
| H | 4 | 2 |
| I | 1 | 9 |
| J | 8 | 1 |
| K | 5 | 1 |
| L | 1 | 4 |
| M | 3 | 2 |
| N | 1 | 6 |
| O | 1 | 8 |
| P | 3 | 2 |
| Q | 10 | 1 |
| R | 1 | 6 |
| S | 1 | 4 |
| T | 1 | 6 |
| U | 1 | 4 |
| V | 4 | 2 |
| W | 4 | 2 |
| X | 8 | 1 |
| Y | 4 | 2 |
| Z | 10 | 1 |
| Blank | 0 | 2 |
Suppose we were to perform an experiment that consisted of drawing one tile from the bag. Let's define the random variable X as the number of points a tile is worth.
a. Explain why X would be considered a discrete random variable
b. What are the possible outcomes for the random variable X?
c. What is the probability that a randomly selected tile is worth 0 points? That is , find P(X=0).
d. What is the probability that a randomly selected tile is worth 4 points? That is , find P(X=4).
e. Similar to (c) and (d), find the proabability for all possible outcomes and list them in the table below.
| Outcomes for X | |||||||
| P(X=x) |
This table is known as the probability distribution of X.
f. Using the table in (e), compute the probability that a tile is worth at most 3 points
g. compute the probability that a tile is worth at least 2 points.
h. find the expected (mean) number of points a Scrabble tile is worth.
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