The Effect of Estimating the Standard Deviation on Measuring Romantic Love

start Fathom and load the file : "EstimatingSD.ftm"

[or load it from the student drive at "G:\\EHS Courses\Intro to Stats\Unit 9\HAGIN EstimatingSD.ftm"]

 

Scenario: A study found that nerve growth factor (NGF) levels in people who recently fell in love is normally distributed with a mean 227 (pg/mL) and a standard deviation 107. Assuming these are true population parameters, we will use Fathom to try to estimate the population mean from a random sample of five people who fell in love.

 

Assumptions: We have learned to construct a 95% confidence interval using the formula x̄ ± z*(σ/√n). In other words, when we use samples to construct 100 confidence intervals, 95 should capture the true value of the population mean.

 

Demonstration :

Part 1: We will start with our true population standard deviation and show how we can use the sample mean of 5 people's NGF levels to try and capture the true population mean. We will press "Collect More Measures" on the Chart (or graph) and count the number of RED bars each time. We should get a sense of the capture rate of the sample means.

 

Part 2: We double-click on the NGF Collection and select the "Measure" tab to see the formula used. We will replace the true population standard deviation with the sample standard deviation to get the formula, x̄ ± z*(s/√n) and demonstrate whether or not we still can capture the true population mean 95 out of 100 times.

 

Part 3: Finally, we will modify the formula x̄ ± z*(s/√n) to use a t-distribution (i.e. x̄ ± t*(s/√n) to actually capture the true population mean 95 out of 100 times. Look up the appropriate t* value in a table or use a calculator. If we do capture the true population mean 95 out of 100 times then we have estimated the true population mean using only the sample mean of 5 people and the sample standard deviation with a 95% confidence interval.

 

Complete :

1. Write appropriate descriptive statistical term for the given symbol.

   Symbol	Statistical Term (i.e. English equivalent)	Fathom Demo Measure (value, variable, or function)
   μ		227
   σ		107
   x̄		sampleMean or mean(NGF)
   s		sampleSD or s(NGF)
   z*		±1.965
   t*		±2.776
   x̄±t*(s/√n)		Lower and Upper

2. Explain the formula: x̄ ± z*(σ/√n)

3. Explain the meaning of one row (grey or red) in the Fathom demo chart.

4. Explain the meaning of the whole Fathom demo chart.

5. Why was random (i.e. randomNormal) used?

6. Explain the statistical term "capture rate"

7. What was the overall purpose of the Fathom demo? In other words, why are we changing the formula x̄ ± z*(σ/√n) to x̄ ± t*(s/√n)?