## statistics and applied probability

### national university of singapore

We organised quizzes for the graduands and the post-graduate students last week and this (the quizzes had only minor differences).  This was to find out how well the curricula were succeeding in teaching the fundamentals of statistics, and also to give students some extra practice before their real exams.  Here are the results (of the graduand quiz: where the postgrad quiz differs is indicated).

1. Here is a function: The derivative with respect to α has no closed form. Explain brieﬂy how you would maximise this function.
You would have to use some numerical method, such as Newton-Raphson or cross entropy.
2. Suppose you have two samples from populations in which binary data are recorded (e.g. liking of the colour pink in science students versus arts students) and wish to assess
whether the two proportions are the same or are different. Why do we use H0 : p1 = p2 and H1 : p1 = p2 and not the other way around?
There are several ways to answer this. To my mind the reason why the null hypothesis is what we assume to be true in carrying out the test is that it is impossible to carry out the test otherwise: there’s no known distribution of the test statistic under the alternative hypothesis, so technically H1 can not be rejected. But there’s an easier answer: unless the data are an exact tie, the likelihood function evaluated at at least one point in the alternative region will be higher than the likelihood evaluated on the constraint that the null is true, so how could the alternative be rejected in favour of the null?
3. Deal two cards from a well-shuffled, French-suited 52-card deck. (a) Are the events {1st is an ace} and {1st is a spade} independent? (b) Are {1st is an ace} and {2nd is a spade} independent?
Both are independent.
4. You have independent samples X1 , . . . , Xn from a normal population with mean µ and standard deviation σ. What is the MLE of (µ, σ)? (But don’t attempt to derive it!) Is this unbiased?
The MLE is (xbar,(n-1)s²/n). It is biased.
5. Which of the following procedures use linear models?
• Two-sample t-test with equal variances
• Paired t-test
• One-way ANOVA
• Two-way ANOVA
• Linear regression
• Multiple regression
• Logistic regression
All except logistic regression are special cases of the linear model.
6. . The sex ratio in most countries is approximately 50:50. A city in such a country has two hospitals, one bigger than the other. The bigger hospital has around 45 births per day, the smaller around 15. The two hospitals keep track of the number of days in the year in which the sex ratio deviates from 50:50 by as much as 60:40 (or 40:60). Which hospital do you think recorded more such days?
• The larger hospital
• The smaller hospital
• About the same (within 5% of each other)
The sampling distribution of the proportion of male births in the smaller hospital has a greater variance because of the central limit theorem.
7. Let the sample mean of a random sample be xbar and the population mean be µ. Which of these is:
• a random variable
• a realisation of a random variable
• a statistic
• a parameter
• an estimator
(You may also answer “both” or “neither” by indicating two yeses or two noes.)
According to frequentist dogma, neither is a random variable, while xbar is a realisation of a random variable, a statistic and an estimator, and µ is a parameter. To a Bayesian, µ is a random variable.
8. A government statistician and an academic economist are investigating the average wage in Singapore (with around 3 million workers). The statistician looks up the wages of
all workers in the country on her governmental data base, while the economist carries out a simple random sample (with replacement) of 200 randomly selected people. Both calculate the mean and standard deviation of wages (respectively (xbar_S , s_S ) and (xbar_E , s_E )). What are the standard errors of the means calculated by the two investigators?
The statistician has a census at her hands, and so there is no uncertainty on the mean: the standard error is 0. The standard error for the economist is s_e/14.14.
9. The Compte de Buﬀon described an experiment, called Buﬀon’s needle experiment. This involves dropping a needle many times over a table marked with parallel lines separated by a distance equal to the needle’s length. It can be shown using geometry and calculus that the probability the needle will touch a line is p = 2/π. Dropping the needle a few hundred times, a mathematician derived a conﬁdence interval for π by manipulating the standard formula for the conﬁdence interval of a proportion, ;
this gave the interval (3.08, 3.18).
(a) What is the probability the true value of the parameter is inside this conﬁdence interval if Zα = 1.96?
100%. π=3.14 which is in (3.08, 3.18).
(b) Why is the value of Zα taken from a normal distribution?
The central limit theorem says that phat is approximately normal for this sample size.
10. Imagine you are going to carry out a two-sample t-test of the null hypothesis that two means are equal. What is the distribution of the resulting p-value under the null hypothesis?
It is uniform on the interval (0,1).

1. Alex

2010/04/23 at 10:16

Ok ok, two criticisms from Adrian:

• it’s not the CLT that makes the smaller hospital have the greater variance in daily proportions, it’s a combination of the CLT and some additional but trivial probabilistic calculations;
• “according to frequentist dogma” is not meant to be read pejoratively. OED provide this as a definition of dogma

The body of opinion formulated or authoritatively stated; systematized belief; tenets or principles collectively; doctrinal system.

This seems to me to be a particularly apt description of both frequentist and Bayesian paradigms, but especially the former, given the way it is usually presented as the “default” or “standard” (or “only”) method of inference in undergraduate classes.

2. So Prof Alex, how were our results?

3. Alex

2010/04/30 at 17:03

Yaoyuan: very mixed!

4. So, did it help in providing ideas to modify the syllabus?

5. YAP Von Bing

2010/05/17 at 13:03

The fact that the smaller hospital has a bigger variance follows from the variance formula for a sum of IID random variables and a change of scale. The CLT is not needed.

6. In 9., formula for confidence inverval of a proportion requires standard deviation and not variance, so p(1-p) must be put under square root sign. Moreover, the confidence interval lenght corresponds to n=40000, which are not so few hundreds but quite a few.
Anyhow, an excellent test.

7. Alex

2010/06/14 at 13:43

Whoops, a bit embarrassing about the missing square root! (Most likely a combination of me misTeXing it and not checking properly, my bad.) The confidence interval length was made up based on my intuition of how broad it should be with a “few hundred” drops: very interesting how badly my intuition failed me, or, put another way, how arrogant I was about how quickly uncertainty would disappear.

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