Business and Economic Statistics

Ages of Groom and Brides- Confidence Intervals with Proportions

We want to estimate the proportion of all marriages in this country for which the bride (i.e., the woman) is younger than the groom (i.e., the man). In the accompanying EXCEL spreadsheet is a sample of 100 couples who applied for marriage licenses in Cumberland Country, Pennsylvania.

(a)Data Cleaning: For how many of these 100 marriages can you determine which partner was younger? (In other words, eliminate the cases in which both bride and groom listed the same age on the marriage license, since you cannot tell which partner is younger in those cases.)

(b) In how many of these marriages is the bride younger than the groom?

(c) Find a 90%, 95% and 99% confidence interval for the proportion of all marriages in this country for which the bride is younger than the groom?

(d) Do any of these intervals include the value 0.5?

(e) Conclude from this sample data whether the bride is younger than the groom in more than half of the marriages in this country.

Measurement Model- Probability Box Model

To what extent is a group of numbers influenced by chance? The main strategy is twofold:

a) Find an analogy between what we are studying (ex, our Dice Lab experimental probabilities) and drawing numbers at random from a box (our theoretical probabilities).

b) Connect the variability you want to know about with the chance variability in the sum of numbers drawn from our box (like our Dice Lab comparison).

The analogy between a chance process and drawing from a box is called a box model, also called a chance model.

In measurement we use what is called a Gauss model for the box model, which states:

Measurement= Exact weight+ Bias+ Error

In the Gauss model, each time a measurement by given instrument is made, a ticket is drawn at random with replacement from the error box. The number on the ticket is the chance error. It is added to the exact value to give the actual measurement. The error distribution is a Gaussian or bell curve.

a) What is the Mean Average of the Gauss model Error box?

-Hint: The standard normal curve.

b) What is the standard deviation of the Gauss model?

-Hint: Is this known?

c) For the readings for machine #1 in Problem #3 below, write the measurement equations.

-Hint: they should be of the form:

“1st measurement= exact weight + 1st draw from error box = x”

Spectrophotometer Recalibration- t-test

In Los Angeles, studies are conducted to find out the levels of CO (carbon monoxide) near freeways with different amounts of traffic. The studies take air samples in special bags, and then measure CO concentrations with a spectrophotometer. This machine measures up to 100 ppm (parts per million by volume) with errors of the order of 10 ppm. They are very delicate, so spectrophotometers have to be calibrated every day- by testing CO levels in a special gas made just for calibration called span gas with CO at 70 ppm.

If the machine reads 70 ppm, then it is ready for use; if not, it has to be adjusted.

The technician takes the following readings on 4 machines using span gas:

#1-71, 68, 79

#2-71, 68, 79, 84,78,85,69

#3-71

#4-71, 84

Use the results of the Gauss model in Problem #2 above as your box model.

In each case, make a t-test to see whether the instrument is properly calibrated.

[a] Show your t-tests

[b] In one case, this is impossible. Which one and why?

Left and Right Handedness― Gender Dominant? – Chi- Square and Hypothesis Testing

Are handedness and gender independent? Let’s look at American age 25-34. Does the distribution of right-handed, left-handed, and ambidextrous―among men, differ from the distribution among the women?

If data were available for every man and woman in the US on handedness, we would have the answer. But there is none．

HANES- the Health and Nutrition Examination Survey of 1976- 1980 – run by the US Public Health Service – looked at a sample of 2,237 Americans age 1 to 74. HANES studied handedness. Here are the results:

Handedness by sex

__________________Men________Women______

Right- handed 934 1,070

Left- handed 113 92

Ambidextrous 20 8

First, convert the table to percentages.

Second, for the base case―the box method in Chi-Square---- Calculate the expected frequency. Hint: start by taking totals in both rows and columns.

Third, state the null and alternative hypothesis.

Fourth, show the formula for Chi-square, and calculate its value. How many degrees of freedom are there?

Fifth, use EXCEL to calculate the Chi-square values.

What is your conclusion?

Farr’s Smallpox Recovery and Death Rates- Observational Inference

William Farr in his article “On prognosis” made epidemiological history by making the first longitudinal analysis of smallpox recovery and death rates tracking the same group over weeks of illness. The table of his analysis is in an EXCEL file attached to the Class Web site.

Using Farr’s data, and create two graphs:

(a) The Population Surviving [in %] vs. the Days Since Onset [i.e., days since start of being sick]. This is called the “Survival Chart”.

(b) The Rate Per 1000 Person- Days vs. the Days Since Onset. On the same graph, show both the deaths and recoveries. Note that mortality rate is defined as the ratio of deaths to infections by the disease under study. This graph is called the “Recovery and Death Rates Chart”.

(c) What inferences do you make from the observations that Farr made?

Chemical Analysis of Unknown Liquid- Sampling

Suppose that you took a drop of liquid from a bottle, for a chemical analysis. If the liquid is well mixed, the chemical composition of the drop should reflect the composition of the whole bottle, and it really wouldn’t matter if the bottle was a test tube or a gallon jug. The chemist doesn’t care whether the drop is 1% or 1/100 of 1% of the solution.

What does this tell you about sample size? Explain.

The Salk Polio Vaccine Field Trials- Experimental Design

In lab trials, the Salk polio vaccine proved safe and caused the production of antibodies against polio. In 1954, the Public Health Service and the National Foundation for Infantile Paralysis (NFIP) were ready to try the vaccine in limited rollout.

The NFIP ran a controlled experiment to test the effectiveness of the vaccine. Children at the age most vulnerable to polio―grades 1, 2 and 3 – were selected from a small number of school districts throughout the country.

Parent permission was required. It was known that higher –income parents would more likely consent to treatment of their children than lower-income parents. Note that polio is a disease of hygiene. Children who live in less hygienic surroundings tend to contract mild cases of polio early in childhood while still protected by antibodies from their mothers. After being infected, the children then generate their own antibodies. So, poorer children have more natural polio antibodies.

Designs for the field trial were proposed, and a number of concerns were raised.

(a) The NFIP originally wanted to vaccinate only grade 2 children whose parents would consent, leaving the children in grades 1 and 3 unvaccinated. Note that polio is a contagious disease. So if by chance, polio hit grade 2 and not grade 1 or 3, then the incidence of polio would have lessened the tested effectiveness of the vaccine.

(b) The next problem was assigning the children to treatment or control groups. Human judgment seemed necessary for this assignment.

(c) Another concern was the possible use of a placebo―an injection of harmless salt in water―in the control group.

(d) Diagnosticians were needed to tell whether the children contracted polio during the experiment. These are many forms of polio which are hard to diagnose. In borderline cases, the diagnosticians could have been influenced by knowing whether the child was vaccinated.

Due the criticisms of the NFIP field test, a second field test was done (Study 2).

Here are the results of two field tests of 1954:

NFIP Study Alternative Study 2

Size Infection Rate Size Infection Rate

Grade 2 (vaccine) 225,000 25 Treatment 200,000 28

Grade 1 and 3 725,000 54 No treatment 200,000 71

Grade 2 (no consent) 125,000 44 No consent 350.000 46

Total 975,000 Total 750,000

Study 2 is randomized controlled double-blind experiment.

(1) Discuss the design issues of the NFIP field test. What are the biases? How should the field study be designed?

(2) Look at the results of Study 2 vs. the NFIP study. Discuss whether there is any bias built into the NFIP study.

(3) Evaluate the success or failure of the NFIP study.

The example (mentioned) above Measurement Model, Dice Lab.

No.09818 Re: Statistics Exam 【青木繁伸】 2009/05/05(Tue) 20:01

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