Dr. Rodrigue, Quant. Methods, Lab 6

Geography 215: QUANTITATIVE METHODS

Dr. Rodrigue

Graded Lab 6: Introduction to Hypothesis Testing

LAB EXERCISE A: Estimation in Sampling

What is the Central Limit Theorem? What does this have to do with inferential statistics?


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What is the conceptual and the calculational difference between the standard error of the mean and the regular standard deviation?


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You have done a systematic spatial sample of 100 quadrats in your 1 sq. km desert study area. Each quadrat is 100 meters square. In each, you have counted every plant identifiable as Artemisia tridentata (desert sagebrush). You have calculated the mean number and standard deviation of these plants per sampled quadrat, in order to estimate the mean number and standard deviation per 100 sq. m throughout your study area (population).
- How many 100 sq. m quadrats could fit in this area? _______________
- What, then, is the sampling fraction? _______________
Your sample mean is 58.12 A. tridentata plant individuals per quadrat, with a sample standard deviation of 5.50 individuals per quadrat. Construct a confidence interval around the mean of 58.12 that will capture the true population mean of plants per quadrat in your entire study area at the 95 percent confidence level.
- Following the advice in M&M, should you incorporate the finite population correction factor in your calculations? Explain your answer.
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- Will you be using the normal table of Z scores or the t table? Explain your choice.
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- Write down the formula you used to construct the interval and then show the values you placed in the formula below.
- Lower bound of the confidence interval (to 2 decimal places of accuracy): _______________
- Upper bound of the confidence interval (to 2 decimal places of accuracy): _______________
For this problem, show your calculations to 2 decimal places of accuracy. As an urban planner in Espalda de Rana, NM, responsible for child-care issues, you want to estimate the mean number of children aged 0-12 per household. The problem is the Census is out of date, creating the need for an estimate based on sampling in the present, but you don't want to spend excessive taxpayer money and staff time on a survey.
What you know so far is there are estimated to be 4,567 households in Espalda de Rana at the present. The mean number of children per household in the 0-12 range back in 1990 was 0.92, with a standard deviation of 0.50. You have estimates from the State that the mean number has declined slightly Statewide about 4 percent. The standard deviation, however, has increased 20 percent (reflecting a national trend toward diversification of household types and the increased number of childless households).
Your job is to figure out the number of households the Planning Department needs to sample in order to create a suitable estimate of the number of these children per household. Your estimate should have a 95 percent confidence of catching the true population mean within 0.25 children of precision (now, there's a concept: What's 0.25 of a kid?).
To do this, you need an estimate of the standard deviation in the population. You've decided to adjust the local numbers for your community by the Statewide changes to create an estimated standard deviation for the formula. The idea behind this is the assumption that the people in Espalda de Rana are just plain folks, regular New Mexicans, and their household formation behavior reflects Statewide trends.
- What is the standard deviation estimate you will use to enter the formula?
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- What is the relevant formula in M&M?
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- What is the proper Z corresponding to your desired confidence level?
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- What is the proper error term, reflecting the acceptable imprecision in your estimated number of children ages 0-12?
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- What is the desirable telephone survey size, then, expressed as an integer rounded up to the next larger 5?
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For this problem, show your values to 2 decimal places of accuracy. Continuing with the offspring of Espalda de Rana, let's say the issue was not the mean number of children 0-12 per household but the proportion of the households having children in this range. Again, you want to have a sample size large enough to create an interval of 0.25 around the estimated proportion at the 95 percent confidence level.
- Now, which of the M&M formulae is the relevant one? _______________
- What is the desirable telephone survey size, then, expressed as an integer rounded up to the next larger 5?
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LAB EXERCISE B: Hypothesis Testing in One Sample Contexts

What is the logical rationale behind the statement of null and alternate hypotheses?


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What is the difference between a directional and a non-directional alternate hypothesis, and how does it relate to number of tails in a test?


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What is the difference between a Type I error and a Type II error and how does a researcher connect them in deciding on a confidence level?


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Two somewhat different approaches to hypothesis testing were presented in class and in the text: the classical approach and the prob-value method. What is the major difference between them?


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Take a look at formulae 8.3 and 8.4 in M&M, for the difference of means t-test. In which way do they differ, and why are they functionally equivalent anyhow?


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For this problem, calculate to 2 decimal places of accuracy, e.g., 0.00. Below are the scores for the population of 215 students turning in the midterm during Fall, 1993.

What is the mean and the median for the following student scores?

          
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               172       136       120       107
               171       136       117       104
               155       133       117       104
               149       131       117        99
               149       131       112        85
               147       128       112        85
               144       128       109        77
               141       123       109
               139       120       107

Calculate the standard deviation (to two decimal places) for the same population distribution:
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Go back to and re-examine the grades for the 215 midterm more closely. You'll notice some of the numbers (8 of them, to be precise) have been set off in boldface. Calculate the basic descriptive statistics for just these eight students (don't forget: to 2 decimal places...):
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Now, not knowing who these folks are, formulate a working hypothesis and the corresponding null hypothesis concerning their being representative of the population from which they come. Use the 95 percent confidence level.
- Working hypothesis: ________________________________________
- Null hypothesis: ________________________________________
Is this going to involve a 1-tailed or 2-tailed test? _______________
Which alpha corresponds to this confidence level? _______________
Looking at the population and the sample size, which test of the difference between the means is going to be the right choice, the Z test or the t-test?
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Why? What's the major procedural difference?


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When you're sure of which test to do and how many tails it should have (and why), consult M & M and do your test. What is the critical statistic for your (my) chosen alpha level?
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Sketch a normal curve and on it label the critical region for rejection of the null hypothesis and the region of non-rejection.

Now, which observed statistic did you calculate for your sample of 8?

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That done, mark the position of your observed statistic on your sketch of the normal curve.

So, what's your decision? to reject the null hypothesis or not to reject the poor little thing?


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 Now, please state what your conclusion MEANS:  

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 What  is the probability that purely random  sampling  would have  
extracted a sample of 8 with a mean so far away from the population mean?
          
                       PROB VALUE _____________________

          

 Okay,  a slight variation.  The Quant Methods Eight are  the graduate  
students.  If you had "known" that ahead of  time, how would your 
decision-making process have been  different, in terms of number of tails and 
critical statistic?  

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 Draw  a second normal curve, showing the new critical statistic  and the 
QM8 observed statistic.          









          
          



first placed on the web:  11/06/98
last revised:  11/06/98
© Dr. Christine M. 
Rodrigue

Geography 215: QUANTITATIVE METHODS

Dr. Rodrigue

Graded Lab 6: Introduction to Hypothesis Testing

first placed on the web: 11/06/98

last revised: 11/06/98

© Dr. Christine M. Rodrigue