In: Statistics and Probability

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.

Null hypothesis: Ho: Die is fair ; each number occur with equal frequency

Alternate hypothesis: Ho: Die is unfair ; at least one number occur with different frequency.

degree of freedom =categories-1= | 5 | |||

for 0.01 level and 5 df :crtiical value X2 = | 15.086 |
from excel:
chiinv(0.01,5) |
||

Decision rule: reject Ho if value of test statistic
X2>15.086 |

applying chi square goodness of fit test: |

relative | observed | Expected | residual | Chi square | |

Category | frequency(p) |
O_{i} |
E_{i}=total*p |
R^{2}_{i}=(O_{i}-E_{i})/√E_{i} |
R^{2}_{i}=(O_{i}-E_{i})^{2}/E_{i} |

1 | 1/6 | 62 | 50.00 | 1.6971 | 2.8800 |

2 | 1/6 | 45 | 50.00 | -0.7071 | 0.5000 |

3 | 1/6 | 63 | 50.00 | 1.8385 | 3.3800 |

4 | 1/6 | 32 | 50.00 | -2.5456 | 6.4800 |

5 | 1/6 | 47 | 50.00 | -0.4243 | 0.1800 |

6 | 1/6 | 51 | 50.00 | 0.1414 | 0.0200 |

total | 1.00 | 300 | 300 | 13.4400 | |

test statistic
X^{2}= |
13.440 |

since test statistic does not falls in rejection region we fail to reject null hypothesis |

we do not have have sufficient evidence at 1% level to conclude that die is unfair, |

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 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 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.

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

If a die is rolled 300 times, use the Chebyshev inequality to
estimate the probability
that the number of occurrences of "three" does not lie strictly
between 45 and 55.

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 fair die is rolled 300 times and each time a number evenly
divisble by three is rolled, a success is recorded. Find the
probability of obtaining the following: Between 90 and 110
successes (inclusive) (Round to four decimal places)

A fair 4-sided die is rolled, let X denote the outcome. After
that, if X = x, then x fair coins are tossed, let Y denote the
number of Tails observed. a) Find P( X >= 3 | Y = 0 ). b) Find
E( X | Y = 2 ). “Hint”: Construct the joint probability
distribution for ( X, Y ) first. Write it in the form of a
rectangular array with x = 1, 2, 3, 4 and...

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)

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