Guide for Mathematics T Assignment, Sem 3/2013

You can't copy all the answers as this semester involve random digits that are impossible for two individuals to have completely same data, hence, this guide will only be assisting you in solving the questions. This serves as only a guide, so no detailed solution here.

Question 1
Give the definition of subjective probability. Find the definition online and describe three examples. Don't just give, describe! For example: "The subjective probability that a China player would win in the match, as deemed by the Chinese player, would be 0.95 because they have faith in him".

Question 2
Generate 30 random digits, and then compartmentalise them into three sections: three different digits, two same digits and three same digits. Tabulate your result, find the respective probability. (If you got 20 out of 30, probability is 0.67).

That is your probabilities for the three categories. Find the answers according to the random numbers you generated.

In making deduction, just say which cases have high chances of occur, and which have low chances.

Question 3
Generate 100 random digits using either excel, calculator, or a website. Here's a website to help you:
http://www.random.org/integers/

Again, compartmentalise them into three categories.

This time you'll be asked to find the 90% and 95% confidence intervals. This involves knowledge on sampling and estimation.

Assuming that the population is normally distributed, to find its 90% confidence intervals, you merely want to find the 90% regions around the centre, which means you'll be eliminating 5% from the left side of the normal distribution, and 5% from the right side of the distribution. Same theory applies to the 95% confidence intervals (find 95% of area under graph at the centre, eliminating 2.5% at the left most and 2.5% at the right most. If you don't get what I mean, search your books for diagrams). Since 95% confidence intervals has a higher confidence, the intervals you get should be larger.

This questions requires only direct application of the formula, so just do it. Show your steps, and make a summary by tabulating them in a table would be helpful.

Question 4
a) Not sure what to do about this part, but we simply write the subjective probability obtained in Q3 in a table.

b)
Generate another 100 random digits, and find the subjective probabilities for each category. Chi-squared Tests is performed to investigate whether to accept or reject the null hypothesis. Your null hypothesis is "the distribution fits the distribution in (a)" and your alternative hypothesis is "the distribution does not fit the distribution in (a)".

You would be expected to have knowledge on Chi-Squared Tests. Just directly apply the formula and see whether the chi-squared calculated exceeds the critical value. If it does, the null hypothesis is rejected, and you can conclude that the distribution obtained in (b) does not fit the distribution obtained in (a).

Note that for chi-squared calculation to be valid, each category must have frequency of at least 5. If your "three same digits" category has less than 5 values, you need to merge it with another category of your choice to form a category of frequency exceeding 5.

Thus, if merging is done, your degree of freedom would be equal to 1 (2 - 1 = 1). You'll have to observe here whether your expected frequency of all categories are bigger or smaller than 10. If your expected frequency is less than 10, then Yates' Correction needs to be employed. But it is unlikely for your expected frequency to be less than 10 now that you've merged two categories. However, if this situation does arise, use this equation:

If all your categories have frequency exceeding 5 and no merging is done, meaning your degree of freedom is 2, and if your degree of freedom is one but all your expected frequency exceeds 10, then ignore this correction and just use the normal chi-squared formula.

Note:
Use Yates' Correction Only When
1) degree of freedom = 1
2) expected frequency < 10
Both conditions need to be met for Yates' Correction to be employed. 

Chi-squared tests is chosen here because the object to be investigated is categorical which does not, and could not, presume a normal distribution graph. (behaviours of categories)

Question 5
(a) Use the subjective probability you obtained earlier.

(b) Generate 64 digits using your calculator. Compartmentalise them again and find the probabilities. Then use hypothesis testing (normal approximation) to check whether to accept or reject the null hypothesis. Your null hypothesis is "the probability that a number has three different digits is more than the probability you have suggested in (a)".

Hypothesis testing of normal approximation is chosen because the object to be investigated here is numerical that could resemble a normal distribution graph. (The probability of three different digits).


*The guide above is only a guide and is not an official answer by MPM.