Learning Outcomes
Lets compare what we have learned about sampling distributions for proportions and for means. Sampling DistributionVariableParameterStatisticCenterSpreadShapeCategorical (example: left-handed or not)p = population proportion[latex]\stackrel{ˆ}{p}[/latex] = sample proportionp[latex]\sqrt{\frac{p(1-p)}{n}}[/latex]Normal if np 10 and n(1 p) 10Quantitative (example: age)μ = population mean, σ = population standard deviation[latex]\overline{x}[/latex] = sample meanμ[latex]\frac{\sigma }{\sqrt{n}}[/latex]Normal if n > 30 (always normal if population is normal)Now we know the conditions that allow us to use a normal model for the sampling distribution of means. As we have done before, we now convert sample means to z-scores and use a standard normal curve to find probabilities and identify unusual sample means. Normal Model Simulation Useful AgainRecall the standard normal model simulation we first used in Probability and Probability Distribution. It was our tool for converting between intervals of z-scores and probabilities. Click here to open this simulation in its own window. ExampleSurprising Heights for Individual Basketball PlayersSuppose we have a population of adult male basketball players and we know their heights: the mean height is μ = 190 cm and the standard deviation of their heights is σ = 7.2 cm. The heights are normally distributed, which is often the case with body measurements. Would it be surprising to find a randomly chosen player from this population with a height of 195 cm? We can answer this question by computing the probability that a randomly chosen player from this population has height greater than 195 cm. To carry out the analysis, lets use X to denote the height of a randomly chosen individual from this population. Since heights are normally distributed, we can convert heights to z-scores and use our simulation to find the probability P(X > 195).
Conclusion: This probability is not very low (almost 25%). We conclude that it would be not be surprising to find a randomly chosen individual from this population with a height of 195 cm. ExampleSurprising Heights for Samples of Basketball PlayersAs before, suppose the heights of individual players are normally distributed with μ = 190 cm and σ = 7.2 cm. Would it be surprising to find a randomly chosen team of 25 players with a mean height of 195 cm? We compute the probability that a random sample of 25 players has a mean height of 195 cm or more. We have to look at the distribution of all sample means for samples of size 25. Heres what we know about this sampling distribution:
Now we can answer this question by computing the probability that a randomly chosen sample of 25 players from this population has mean height greater than 195 cm. To carry out the analysis, lets use [latex]\overline{X}[/latex] to denote the mean height of a random sample of 25 players from this population. Because mean heights are normally distributed, we can convert mean heights to z-scores and use our simulation to find the probability P( [latex]\overline{X}[/latex] > 195).
Conclusion: This probability is very low (much, much less than 1%). We conclude that it would be very surprising to find a random sample of 25 players from this population with a mean height of 195 cm. Its interesting to notice that the height cutoff we used in these two examples is the same (195 cm). When considering the individual, we concluded that finding a randomly chosen individual with height of 195 cm would not be surprising. However, when we considered the team, we concluded that it would be very surprising to find a random sample of 25 players with a mean height of 195 cm. This makes sense because as sample size grows, variability shrinks (here we considered a sample of size 1 versus a sample of size 25). Click here to open the normal simulation in a separate window to answer the following questions. Try ItThe annual salary of teachers in a certain state X has a mean of μ = $54,000 and standard deviation of σ = $5,000. What Have We Learned Here?We need to be careful before using the normal model to find probabilities associated with sample means.
Note: The logic of inference in this module is familiar. We make a claim about a population mean. We use a random sample to test our claim. We determine whether it is probable that random samples have means as extreme as the actual sample. If this is very unlikely, then we conclude this sample probably could not have come from this population and that the claim about the population mean is probably false. We used logic like this in Modules 7, 8, and 9 in the context of proportions. In this module, we further develop this idea in the context of means. Lets Summarize
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