. What is the difference between the null and alternative hypotheses statements in one-tailed and two-tailed t

What is the difference between the null and alternative hypotheses statements in one-tailed and two-tailed tests? How can manufacturing companies use the standard normal distribution to determine quality control of their products?





2. What is the "perfect" standard normal distribution? Explain your answer. What value is business research and hypothesis testing to a company?|||Consider the hypothesis as a trial against the null hypothesis. the data is evidence against the mean. you assume the mean is true and try to prove that it is not true.





If the question statement asks you to determine if there is a difference between the statistic and a value, then you have a two tail test, the null hypothesis, for example, would be 渭 = d vs the alternate hypothesis 渭 鈮?d





if the question ask to test for an inequality you make sure that your results will be worth while. for example. say you have a steel bar that will be used in a construction project. if the bar can support a load of 100,000 psi then you'll use the bar, if it cannot then you will not use the bar.





if the null was 渭 鈮?100,000 vs the alternate 渭 %26lt; 100,000 then will will have a meaningless test. in this case if you reject the null hypothesis you will conclude that the alternate hypothesis is true and the mean load the bar can support is less than 100,000 psi and you will not be able to use the bar. However, if you fail to reject the null then you will conclude it is plausible the mean is greater than or equal to 100,000. You cannot ever conclude that the null is true. as a result you should not use the bar because you do not have proof that the mean strength is high enough.





if the null was 渭 鈮?100,000 vs. the alternate 渭 %26gt; 100,000 and you reject the null then you conclude the alternate is true and the bar is strong enough; if you fail to reject it is plausible the bar is not strong enough, so you don't use it. in this case you have a meaningful result.





Any time you are defining the hypothesis test you need to consider whether or not the results will be meaningful.





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You can use the standard normal in quality control because of the central limit theorem.





Let X1, X2, ... , Xn be a simple random sample from a population with mean 渭 and variance 蟽虏.





Let Xbar be the sample mean = 1/n * 鈭慩i


Let Sn be the sum of sample observations: Sn = 鈭慩i





then, if n is sufficiently large:





Xbar has the normal distribution with mean 渭 and variance 蟽虏 / n


Xbar ~ Normal(渭 , 蟽虏 / n)





Sn has the normal distribution with mean n渭 and variance n蟽虏


Sn ~ Normal(n渭 , n蟽虏)





The great thing is that it does not matter what the under lying distribution is, the central limit theorem holds. It was proven by Markov using continuing fractions.





if the sample comes from a uniform distribution the sufficient sample size is as small as 12


if the sample comes from an exponential distribution the sufficient sample size could be several hundred to several thousand.





if the data comes from a normal distribution to start with then any sample size is sufficient.


for n %26lt; 30, if the sample is from a normal distribution we use the Student t statistic to estimate the distribution. We do this because the Student t takes into account the uncertainty in the estimate for the standard deviation.


if we now the population standard deviation then we can use the z statistic from the beginning.


the value of 30 was empirically defined because at around that sample size, the quantiles of the student t are very close the quantiles of the standard normal.





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The perfect standard normal is Normal(渭 = 0, 蟽虏 = 1).





as n 鈫?鈭? Xbar ~ Normal(渭x , 蟽x虏 / n)





and





(Xbar - 渭x ) / sqrt( 蟽x虏 / n) ~ Normal(渭 = 0, 蟽虏 = 1)





because of this, in a sample of sufficient size we can approximate the behavior of the mean with the normal distribution which is easily translated into the standard normal. This is the basis for nearlly all parametric hypothesis testing.