In: Statistics and Probability

A die is rolled 50 times and the following are the outputs:

6 1 3 4 2 6 3 5 1 3 6 1 6 6 3 3 6 5 2 4 1 6 5 3 1 2 5 2 1 2 4 1 4 1 5 5 6 6 2 1 1 2 5 6 5 5 6 3 1 3

What is the p-value of the chi-square test that the die is unbiased?

Solution:

Test statistic formula for chi square test is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

The calculation table for the above test statistic is given as below:

Outcome |
O |
E |
(O - E)^2/E |

1 |
11 |
8.333333 |
0.853333333 |

2 |
7 |
8.333333 |
0.213333333 |

3 |
8 |
8.333333 |
0.013333333 |

4 |
4 |
8.333333 |
2.253333333 |

5 |
9 |
8.333333 |
0.053333333 |

6 |
11 |
8.333333 |
0.853333333 |

Total |
50 |
50 |
4.24 |

Chi square = ∑[(O – E)^2/E]

Chi square = 4.24

Test statistic = 4.24

We are given

N = 6 outcomes

Degrees of freedom = df = N – 1 = 6 – 1 = 5

So, required p-value for this test by using chi square table or excel is given as below:

P-value = 0.515404429

**P-value =** **0.5154**

P-value > α = 0.05

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that die is unbiased.

A die was rolled 300 times. The following frequencies were
recorded.
Outcome 1 2 3 4 5 6
Frequency 62 45 63 32 47 51
Do these data indicate that the die is unfair ? Use a 1% level
of significance.

A die is rolled 60 times with the following results for the
outcomes 1, 2, 3, 4, 5, and 6, respectively: 13, 7, 6, 11, 10, and
13.
Specify the null and alternative hypotheses.
Use Minitab to perform the analysis. What is the
conclusion?
Show how to obtain by hand the found in the Minitab
output.
Demonstrate how to find the bounds on the p-value using a
table

A die is rolled 30 times. The outcomes are shown below:
1 1 2 3
3 6 6 5
3 2 1 3
5 6 3 3
3 2 1 4
2 4 1 5
6 1 3 4
2 3
a. Complete the table:
Outcomes
Class Boundaries
Freq.
Cf
Rf
%f
Central angle
1
2
3
4
5
6
Total

A six sided die is rolled 4 times. The number of 2's rolled is
counted as a success.
Construct a probability distribution for the random
variable.
# of 2's
P(X)
Would this be considered a binomial random variable?
What is the probability that you will roll a
die 4 times and get a 2 only once?
d) Is it unusual to get no 2s when rolling a die 4 times? Why or
why not? Use probabilities to explain.

A six sided die is rolled 4 times. The number of 2's rolled is
counted as a success.
Construct a probability distribution for the random
variable.
# of 2's
P(X)
Would this be considered a binomial random variable?
What is the probability that you will roll a
die 4 times and get a 2 only once?
d Is it unusual to get no 2s when
rolling a die 4 times? Why or why not? Use probabilities to
explain.

Assume that a fair die is rolled. The sample space is
, 1, 2, 3, 4, 56
, and all the outcomes are equally likely. Find
P
Greater than 4
. Write your answer as a fraction or whole number.
Assume that a fair die is rolled. The sample space is
, 1, 2, 3, 4, 56
, and all the outcomes are equally likely. Find
P
Greater than 4
. Write your answer as a fraction or whole number.

a die is rolled 6 times let X denote the number of 2's that
appear on the die.
1. show that X is binomial.
2. what is the porbaility of getting at least one 2.
3. find the mean and the standard deviaion of X

a fair 6-sided die is rolled three times. What is the
probability that all three rolls are 2? Round the answer to four
decimal places

Example 4: A fair six-sided die is rolled six times. If
the face numbered k is the outcome on roll k for k = 1, 2, 3, 4, 5,
6 we say that a match has occurred. The experiment is called a
success if at least one match occurs during the six trials.
Otherwise, the experiment is called a failure. The outcome space is
O = {success, failure}. Let event A = {success}. Which value has
P(A)?
**This question has...

Part 1: Two 6-sided dice are rolled. One die is
a standard die, and the other is a Fibonacci die with sides 1, 1,
2, 3, 5, and 8. Submit a file with the probability distribution for
this experiment.
Part 2: Two 6-sided dice are rolled. One die is
a standard die, and the other is a Fibonacci die with sides 1, 1,
2, 3, 5, and 8. What is the shape of the probability
distribution?
Part 3: Two 6-sided...

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