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. After finding the test statistic and p-value, if the p-value is less than or equal to the significance level of the test we reject the null and conclude the alternate hypothesis is true. If the p-value is greater than the significance level then we fail to reject the null hypothesis and conclude it is plausible. Note that we cannot conclude the null hypothesis is true, just that it is plausible.
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 μ = d vs the alternate hypothesis μ ≠ 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 μ ≥ 100,000 vs the alternate μ < 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 μ ≤ 100,000 vs. the alternate μ > 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.
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: μ ≤ Δ or
H0: μ ≥ Δ or
H0: μ = Δ
Find the test statistic t = (xbar - Δ ) / (sx / √ (n))
where xbar is the sample average
sx is the sample standard deviation, if you know the population standard deviation, σ , then replace sx with σ 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 > 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: μ > Δ; p-value is the area to the right of t
H1: μ < Δ; p-value is the area to the left of t
H1: μ ≠ Δ; p-value is the area in the tails greater than |t|
If the p-value is less than or equal to the significance level α, i.e., p-value ≤ α, 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 > α, 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: μ ≤ 1750 vs. H1: μ > 1750
The test statistic is:
t = ( 2017 - 1750 ) / ( 867 / √ ( 20 ))
t = 1.377232
The p-value = P( t_ 19 > t )
= P( t_ 19 > 1.377232 )
= 0.09222523
Since the p-value is greater than the significance level of 0.025 we fail to reject the null hypothesis and conclude μ ≤ 1750 is plausible.