Question

Describe your research question, and explain its importance. Describe how you would use the four-step hypothesis test process to answer your research question. Explain how using a t test could help you answer your research question.

Answer 1

See explanation below

Explanation:

Data given and notation

First we need to find the sample mean and deviation from the data with the following formulas:

`\bar X =(\sum_(i=1)^n X_i)/(n)`

`s=\sqrt{(\sum_(i=1)^n (X_i -\bar X)^2)/(n-1)}`

`\bar X` represent the sample mean

`s` represent the sample standard deviation

`n` sample size

`\mu_o` represent the value that we want to test

`\alpha` represent the significance level for the hypothesis test.

z would represent the statistic (variable of interest)

`p_v` represent the p value for the test (variable of interest)

State the null and alternative hypotheses.

We have three possible options for the null and the alternative hypothesis:

Case Bilateral

Null hypothesis:
`\mu = \mu_o`

Alternative hypothesis:
`\mu \\eq \mu_o`

Case Right tailed

Null hypothesis:
`\mu \leq \mu_o`

Alternative hypothesis:
`\mu > \mu_o`

Case Left tailed

Null hypothesis:
`\mu \geq \mu_o`

Alternative hypothesis:
`\mu < \mu_o`

We assume that w don't know the population deviation, so for this case is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (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

We can replace in formula (1) and the value obtained is assumed as
`t_o`

Calculate the P-value

First we need to find the degrees of freedom:

`df=n-1`

Case two tailed

Since is a two-sided tailed test the p value would be:

`p_v =2*P(t_(df)>|t_o|)`

Case Right tailed

Since is a one-side right tailed test the p value would be:

`p_v =P(t_(df)>t_o)`

Case Left tailed

Since is a one-side left tailed test the p value would be:

`p_v =P(t_(df)<t_o)`

Conclusion

The rule of decision is this one:

`p_v >\alpha` We fail to reject the null hypothesis at the significance level
`\alpha` assumed

`p_v <\alpha` We reject the null hypothesis at the significance level
`\alpha` assumed