##### Question

In: Statistics and Probability

# A die was rolled 300 times. The following frequencies were recorded. Outcome 1 2 3 4...

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.

## Solutions

##### Expert Solution

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) Oi Ei=total*p R2i=(Oi-Ei)/√Ei R2i=(Oi-Ei)2/Ei 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 X2= 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,

## Related Solutions

##### A die is rolled 50 times and the following are the outputs: 6 1 3 4...
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,...
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 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 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  ...
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...
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,...
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...
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 =...
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...
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)