A sample mean, sample standard deviation, and sample size are given. Perform the required hypothesis test about the mean, μ, of the normal population from which the sample was drawn. , , n = 9, H0: μ ≤ 2.85, Ha: μ > 2.85, α = 0.01 Test statistic: t = 2.24. Critical value: t = 2.821. Do not reject H0. There is not sufficient evidence to support the claim that the mean is greater than 2.85. Test statistic: t = 2.24. Critical value: t = 2.33. Do not reject H0. There is not sufficient evidence to support the claim that the mean is greater than 2.85. Test statistic: t = 2.24. Critical value: t = 2.896. Reject H0. There is sufficient evidence to support the claim that the mean is greater than 2.85. Test statistic: t = 2.24. Critical value: t = 2.896. Do not reject H0. There is not sufficient evidence to support the claim that the mean is greater than 2.85.

The statistic on this case is given by

The degrees of freedom are given by:

Since we have a right tailed test we need to find a critical value on the t distribution with 8 df who accumulates 0.01 of the area on the right and we got:

Since the calculated value is lower than the critical value we FAIL to reject the null hypothesis. And the best solution would be:

Test statistic: t = 2.24. Critical value: t = 2.896. Do not reject H0. There is not sufficient evidence to support the claim that the mean is greater than 2.85.

Step-by-step explanation:

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is higher than 2.85, the system of hypothesis would be:  

Null hypothesis:  

Alternative hypothesis:  

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

 (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

The statistic on this case is given by

Critical value

The degrees of freedom are given by:

Since we have a right tailed test we need to find a critical value on the t distribution with 8 df who accumulates 0.01 of the area on the right and we got:

Decision

Since the calculated value is lower than the critical value we FAIL to reject the null hypothesis. And the best solution would be:

Test statistic: t = 2.24. Critical value: t = 2.896. Do not reject H0. There is not sufficient evidence to support the claim that the mean is greater than 2.85.

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this