What is the probability of observing this sample if the hypothesis is true?

You know that the random variable X is distributed normally, but you are uncertain of its true mean or variance. You hypothesize that the true mean is 20. You collect a sample: 20,32,36,26 .What is the probability of observing this sample if the hypothesis is true?
Small Sample Hypothesis Test for mean:





In order for this test to be valid the data must come from a normal population. If this is not the case then this test is not valid and other methods, such as a randomization test or permutation test should be used.





Assuming the normality assumption is valid to test the null hypothesis





H0: 渭 鈮?螖 or


H0: 渭 鈮?螖 or


H0: 渭 = 螖


Find the test statistic t = (xbar - 螖 ) / (sx / 鈭?(n))





where xbar is the sample average


sx is the sample standard deviation, if you know the population standard deviation, 蟽 , then replace sx with 蟽 in the equation for the test statistic.


n is the sample size





and t follows the Student t distribution with n - 1 degrees of freedom. We use the Student t distribution to account for the uncertainty in the estimate of the variance.


As the degrees of freedom approach infinity the Student t converges in probability to the Standard Normal. In most cases the values of the percentiles of the Student t are close enough to the Standard Normal when the degrees of freedom are greater than 30. This is the source of the empirical rule of thumb that samples of size %26gt; 30 have a mean that is normally distributed. Keep that in mind as well, for these hypothesis tests we are assuming the mean is normally distributed. This assumption is easy to verify if the data is normally distributed. The Central Limit Theorem accounts of all other means.





The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.





H1: 渭 %26gt; 螖; p-value is the area to the right of t


H1: 渭 %26lt; 螖; p-value is the area to the left of t


H1: 渭 鈮?螖; p-value is the area in the tails greater than |t|





If the p-value is less than or equal to the significance level 伪, i.e., p-value 鈮?伪, then we reject the null hypothesis and conclude the alternate hypothesis is true.





If the p-value is greater than the significance level, i.e., p-value %26gt; 伪, the significance level then we fail to reject the null hypothesis and conclude that the null is plausible. Note that we can conclude the alternate is true, but we cannot conclude the null is true, only that it is plausible.





The hypothesis test in this question is:





H0: 渭 = 20 vs. H1: 渭 鈮?20





The test statistic is:


t = ( 28.5 - 20 ) / ( 7 / 鈭?( 4 ))


t = 2.428571





The p-value = P( t_ 3 %26gt; |t| )


= P( t_ 3 %26lt; -2.428571 ) - P( t_ 3 %26gt; 2.428571 )


= 2 * P( t_ 3 %26lt; -2.428571 )


= 0.0934493





Since the p-value is greater than the significance level, i used 5% for the significance level, we fail to reject the null hypothesis and conclude 渭 = 20 is plausible.