If a dice is rolled one time, classical probability would indicate that the probability of a "two" should be 1/6. If the dice is rolled 60 times and comes up "two" only 9 times does that suguest that the dice is loaded? why or why not
A hypothesis related to the population proportion will be tested. In this case,
Theoretical proportion, π0 = 1/6,
number of observations, n = 60
Observed proportion, p = 9/n = 9/60
In the case where n>30, nπ0>5 and n(1-π0)>5, the normal distribution could be used as an approximation for the test of the null hypothesis, H0, where
H0 : the sample proportion, π = 1/6.
The sample standard error, σp
=sqrt(π0(1-π0)/n)
=sqrt((1/6)(5/6)/60)
=0.048
z=(p-π0)/σp
=(9/60-1/6)/0.048
=-0.347
critical z (α=0.05)=-1.645 (from the Normal distribution table).
Since -0.347 is greater than -1.645, the null hypothesis is not rejected at the 5% level of significance, i.e. there is no indication that the die is loaded.
Find the chance that if you toss a pair of dice, you get 6 for the sum.
To determine whether the dice is loaded or not, we can use the concept of empirical probability. Empirical probability refers to conducting a probability experiment (such as rolling a dice in this case) multiple times and then analyzing the results.
In this scenario, if a fair, unbiased dice is rolled 60 times, we would expect the number of times a "two" is rolled to be approximately (60 * 1/6) = 10. According to classical probability, the probability of rolling a "two" on a fair dice is indeed 1/6.
However, we need to compare the expected value (10) with the observed value (9). In this case, since 9 is relatively close to the expected value of 10, it does not strongly suggest that the dice is loaded. There is a possibility that this deviation is due to random chance or inherent variability in the outcomes.
To gain more confidence in our conclusion, we can perform a hypothesis test. The null hypothesis (H0) would state that the dice is fair and unbiased, while the alternative hypothesis (Ha) would state that the dice is loaded. By calculating the appropriate statistical test (e.g., chi-square test), we can determine whether the observed result significantly deviates from what we would expect with a fair dice. If the p-value associated with the hypothesis test is below a predetermined significance level (often 0.05), we can reject the null hypothesis and conclude that the dice may indeed be loaded.
In summary, based solely on the observation of rolling a "two" nine times out of 60 rolls, it does not strongly suggest that the dice is loaded. Further statistical analysis, such as a hypothesis test, is needed to draw a more definitive conclusion.