Statistical Investigation - Lecture Notes | STAT 2, Study notes of Statistics

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STAT 2

Lecture 34:

Statistical investigations

● I want to know if the political views of Berkeley students are related to their place of birth. How do I go about studying this?

● I decide to do a chi-square test for independence ● Null hypothesis is that place of birth and political views are independent ● Alternative is that they're not independent

Observed Calif USA Overseas Total

• W hig
• Fed
• Other
• Total
• W hig 50 33.33 16.67 Expected Calif USA Overseas Total
• Fed 50 33.33 16.67
• Other 50 33.33 16.67
• Total
• W hig 2 0.33 2. (O-E)^2/E Calif USA Overseas
• Fed 2 8.33 2.
• Other 0 5.33 10.
• Chi-square statistic:

● A scientist has developed a pill he thinks causes weight loss. He wants me to run a trial to see if the pill works. How should I set up such a trial?

One possible design (not necessarily the best): ● Get a group of 200 volunteers ● Randomly split them into a group that gets the treatment and a group that gets the placebo (also double-blind) ● Measure their change in weight after a fixed length of time (“after” – “before”) How would we analyse the results?

● Null hypothesis: average of all treatment numbers equals average of all control numbers ● Alternative: average of all treatment numbers is less than average of all control numbers

Let's say these are the observed changes after three months: ● Treatment group of 100: average change was -2.4 pounds, with an SD of 8 pounds ● Control group of 100: average change was -2.0 pounds, with an SD of 8 pounds

● z -statistic = (-0.4 – 0)/1.13 = -0. ● P -value = P(Z < -0.35) = 36% The difference between the performance of the treatment group and the control group is explainable by chance variation. We don't have evidence that the diet pill works. (That doesn't mean it doesn't work.)

● I need to test a checkweight to see if it really weighs 1 kg. I have a scale that I know to be unbiased and to have normally distributed errors, following the Gauss model. I only have time to make ten measurements. How should I perform the test?

Aside: why SD+? ● When we're using the sample standard deviation to estimate the population/box standard deviation, as we do in most tests, it's better to use the SD+ rather than the SD. This only really matters for small samples. (Of course, if we knew the true box SD it would be even better to use that, but we usually don't.)

● The ten measurements have a mean of 1.001 kg and an SD+ of 0.008 kg ● Expected weight under null = 1 kg ● SE of average of measurements = 0.008/sqrt(10) = 0.0025 kg

● In the Mega Millions, one of 46 Megaballs numbered 1 to 46 is drawn at random from a rotating drum. You are hired to check that each ball is equally likely. How will you do this?

● Since hundreds of millions of dollars are at stake, you decide to make 10000 draws with replacement ● Null hypothesis is that all numbers are equally likely ● Alternative is that they're not