Estimation Practise + L7 Summary

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Summary Chapter L7 - Sample Means

For example Klout Scores with N= 1048, mean=37.72 and sigma = 16.04

When taking any sample x from the population 1048, the mean will equal population mean

For the sampling error with e.g. n = 10, the sample mean standard deviation can be calculated via: sigma/n^0.5 (standard error)

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Mean of Treated Populations

In case everyone from the cloud population would use the Bieber tweeter, the mean Klout Score should shift to 40 from 37.71 based on the one sample 40

ONE sample mean of 40 does not mean the POPULATION MEAN = 40 on the dot

1 Standard Deviation = 68% of values/probability of 68% that values are selected

2 Standard Deviation in Sampling Distribution = 2x sigma / n ^0.5 = 95% of sample means will fall into this bracket

2 sigma / sqrt(n) = margin of error = 95% of sample means within

In order to calculate the interval bounds for the new population estimation, the initial mean & standard deviation is needed as well as n

N = 40 | SD = 2.71 | --> 16,04*2.71 / 40^0.5 | 2xSE = 2,71*2 = 5,42

Confidence Interval Bounds: 45,42 > 40 > 34,58 ----> 95% Confidence Interval

Sampling Distribution

95% of means in the sampling distribution fall within -1.96 to +1.96 SE's (standard errors)

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Calculation of exact Z scores

95% interval if everyone used the Bieber Tweeter:

Mean = 40 & SD = 2.71

2.71 * 1,96 = +-5,31

Confidence interval 95% = (34,69 < 40 < 45,31)

Interval estimate for the new population

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Causation = The bigger the sample, the smaller the confidence interval around the mean

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Calculation of new z score within -1.98 to 1.98 confidence interval

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Generalization of CI calculations

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Calculate margin of error

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Sample Errors for Engagement and Learning Score

Z Score

Z Probability of 8.94 or 8.35 occurring

The higher the z score post-experiment, the higher the chances of change having an effect on the distribution.

In this example the chance of 0.000000001 in measure of engagement clearly shows that under normal circumstances this value could not have been reached. Post-song however, this score was reached and clearly shows an increased engagement.

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Summary

Chapter of estimation:

Clear picture of how changes in sample means provide information about the probability structure of the population

E.g. shift of sample mean of Klout Scores to 40 by using the Bieber Tweeter or increase of the engagement score to 8.94 for a population of 20 by adding song about hypothesis testing

Process of calculation from mean, standard deviation to sample means, sample errors, z scores and z score probabilities got a lot clearer

Mean = Sum of values / n

Standard deviation = Sum of (xi - xmean)^2 = avg. of xi-xmean^2 = variance = sqrt(variance) = standard deviation ---> Distance for 1 sigma from the mean (probabilities can be calculated via z table or calculus)

Sample Mean = Population Mean

Sampling Error = Sigma / sqrt(n of sample)

Z score = xi - sm / se

Z probabilities = z score table