Question

a.Explain intuitively what does population mean and variance tells you about the distribution of a random variable?

b.Can you think of a reason why do we mostly only care about mean and variance a random variable since the distribution function could involve more than the two parameters?

c.Explain why we could use a sample data to draw conclusions about a population parameter?

d.Is p-value of a hypothesis test a random variable, why?

e.If you reject your null hypothesis, you have proved that the null hypothesis is wrong. Is this a correct statement, why?

Answer 1

a.Explain intuitively what does population mean and variance tells you about the distribution of a random variable?

answer - Population mean is the overall average of the Whatever data we have to select, that means Population mean is based on all values used in the data. Population Variance is the variation in the data we have been selected and Population variance is also based on all values in the data.

For example 1) Suppose we consider Poisson Distribution X ~ Poisson() here is the Parameter of the Poisson distribution, In poisson distribution

Population mean = ​​​​​​​ and Population variance = ​​​​​​​ that is both values depend on overall data.

b.Can you think of a reason why do we mostly only care about mean and variance a random variable since the distribution function could involve more than the two parameters?

Answer - Mean and Variance is the single values which represents overall idea about the data, So we need to choose sufficient data for finding the values of mean and variance. Some Distribution involved more than two parameter in the distribution function.

For example, Normal and lognormal distributions involve more than two parameter in their distribution function.

c.Explain why we could use a sample data to draw conclusions about a population parameter?

Answer - We have to choose a sample data for finding because if Population is very large then we have select a small part of population and find sample mean and sample variance for the chosen sample data and we have find some conclusions about the population, because the very large data are time consuming and saving money and manpower we better to use sample from those population and calculate value of we need.

d.Is p-value of a hypothesis test a random variable, why?

Answer - The p-value approach to hypothesis testing uses the calculated probability to determine whether there is evidence to reject the null hypothesis. That is in our data we have to use 5 % level of significance that is = 0.05

then 1) If P-value then we accept null hypothesis

2) If P-value then we reject null hypothesis.

e.If you reject your null hypothesis, you have proved that the null hypothesis is wrong. Is this a correct statement, why?

Amswer - We set null and alternative hypothesis for Population Parameter that is the Population data, and when we choose small part of population and carry out the testing. The reject or accept the null hypothesis is purely based on selected sample data only, it does not means if one of chosen sample had reject null hypothesis that means it rejects all samples we have to taken. If we choose different samples from the data it may possible to accept or reject the null hypothesis. That means Reject null hypothesis means selected sample values does not better fit for given Population data.