In: Statistics and Probability

- 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

Null Hypothesis: die is Fair

alternate hypothesis: die is not fair

---------

Minitab output-

since, p value >α=0.05, fail to reject Ho

conclusion: die is fair

------------------

observed frequencey, O | expected proportion | expected frequency,E | (O-E) | (O-E)² | (O-E)²/E |

13 | 0.17 | 10.000 | 3.00 | 9.00 | 0.900 |

7 | 0.17 | 10.000 | -3.00 | 9.00 | 0.900 |

6 | 0.1667 | 10.000 | -4.00 | 16.00 | 1.600 |

11 | 0.1667 | 10.000 | 1.00 | 1.00 | 0.100 |

10 | 0.1667 | 10.000 | 0.00 | 0.00 | 0.000 |

13 | 0.1667 | 10.000 | 3.00 | 9.00 | 0.900 |

chi square test statistic,X² = Σ(O-E)²/E =
4.4000

level of significance, α= 0.05

Degree of freedom=k-1= 6 -
1 = 5

P value = 0.4934 [ excel function:
=chisq.dist.rt(test-stat,df) ]

Decision: P value >α , Do not reject Ho

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 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?

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 balanced die with six sides is rolled 60 times. For the
binomial distubtion of X = Numbers of 6’s, what is n, what is p,
and what is q Find the mean and standard deviation of the
distribution of X. What do these values tell you? If you observe X
= 2, would you be skeptical that the die is balanced? Explained
why, based on the mean and standard deviation of . How long would
the value of X...

You roll a die 48 times with the following results.
Number
1
2
3
4
5
6
Frequency
3
1
15
13
4
12
Use a significance level of 0.05 to test the claim that the die is
fair.
(PLEASE SHOW ALL YOUR WORK)

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 three-sided fair die with faces numbered 1, 2 and 3 is rolled
twice. List the sample space. S =
b.{ List the following events and their probabilities. Write
probabilities in non-reduced fractional form A = rolling doubles =
{ P(A)= / B = rolling a sum of 4 = { P(B)= / C = rolling a sum of 5
= { P(C)=
C. Are the events A and B mutually exclusive? If yes, why? If
not, why not?
D.Are...

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