In: Statistics and Probability

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)**

**SOLUTION:**

From given data,

Doing an observed/expected value table,

O |
E |
(O-E)^{2} /
E |

3 | 8 | (3-8)^{2} / 8 = 3.125 |

1 | 8 | (1-8)^{2} / 8 = 6.125 |

15 | 8 | (15-8)^{2} / 8 = 6.125 |

13 | 8 | (13-8)^{2} / 8 = 3.125 |

4 | 8 | (4-8)^{2} / 8 = 2 |

12 | 8 | (12-8)^{2} / 8 = 2 |

Using ,

=

= 3.125+6.125+6.125+3.125+2+2

= 22.5

**Degree of freedom**

** = 6**

df = ** -1** = 6-1 =
5

significance level = = 0.05

**Critical value**

= 11.07049769

Also, the p value is

p = 0.000421

Thus, comparing and
[or, p and significance level], we REJECT THE NULL HYPOTHESIS.

There is significant evidence at 0.05 level that the die is not fair.

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

You roll a balanced die two times. Is the second roll independent of the first roll?

You roll a die, winning nothing if the number of spots is
even, $3 for a 1 or a 3, and $18 for a 5.
a) Find the expected value and standard deviation of your
prospective winnings.
b) You play three times. Find the mean and standard deviation
of your total winnings.
c) You play 30 times. What is the probability that you win at
least $160?

1) Imagine you roll a fair 4 sided dice 4 times.
what is the number of possible outcomes as a sum of successes?
2) What is the probability you will roll 3 successes,
if the result of a 4 is considered a success?
3) what is the probability you will roll 3 or 4 successes, of the
results of a 4 is considered a success?
4) what is the probability you will roll 3 or 4
successes. of the result...

You roll a fair die three times. What is the probability of each
of the following?
a) You roll all 4's.
b) You roll all even numbers.
c) None of your rolls gets a number divisible by 3.
d) You roll at least one 2.
e) The numbers you roll are not all 2's.

Suppose that you roll a die and your score is the
number shown on the die. On the other
hand, suppose that your friend rolls five dice and his score is the
number of 6’s shown out of five rollings. Compute the
probability
(a) that the two scores are equal.
(b) that your friend’s score is strictly smaller than yours.

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 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?

Assume we roll a fair four-sided die marked with 1, 2, 3 and
4.
(a) Find the probability that the outcome 1 is first observed after
5 rolls.
(b) Find the expected number of rolls until outcomes 1 and 2 are
both observed.
(c) Find the expected number of rolls until the outcome 3 is
observed three times.
(d) Find the probability that the outcome 3 is observed exactly
three times in 10 rolls
given that it is first observed...

PROJECT C
Roll a 6-sided die 800 times. If you do not desire to roll a
die manually, STATDISK can be used to simulate the process. To use
STATDISK, go to “Data” at the top of the STATDISK window, and then
choose “Dice Generator”. The “Dice Roll Random Sample Generator”
window will appear. Then in that
window, put 800 in for “Sample Size”, put 1 in for “Num Dice”,
and put 6 in for “Num Sides”. Please disregard the “Random...

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