Respuesta :
Answer:
See explanation below
Step-by-step explanation:
Data given and notation
First we need to find the sample mean and deviation from the data with the following formulas:
[tex]\bar X =\frac{\sum_{i=1}^n X_i}{n}[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]
[tex]\bar X[/tex] represent the sample mean
[tex]s[/tex] represent the sample standard deviation
[tex]n[/tex] sample size
[tex]\mu_o [/tex] represent the value that we want to test
[tex]\alpha[/tex] represent the significance level for the hypothesis test.
z would represent the statistic (variable of interest)
[tex]p_v[/tex] 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:[tex]\mu = \mu_o[/tex]
Alternative hypothesis:[tex]\mu \neq \mu_o[/tex]
Case Right tailed
Null hypothesis:[tex]\mu \leq \mu_o[/tex]
Alternative hypothesis:[tex]\mu > \mu_o[/tex]
Case Left tailed
Null hypothesis:[tex]\mu \geq \mu_o[/tex]
Alternative hypothesis:[tex]\mu < \mu_o[/tex]
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:
[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (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 [tex]t_o[/tex]
Calculate the P-value
First we need to find the degrees of freedom:
[tex] df=n-1[/tex]
Case two tailed
Since is a two-sided tailed test the p value would be:
[tex]p_v =2*P(t_{df}>|t_o|)[/tex]
Case Right tailed
Since is a one-side right tailed test the p value would be:
[tex]p_v =P(t_{df}>t_o)[/tex]
Case Left tailed
Since is a one-side left tailed test the p value would be:
[tex]p_v =P(t_{df}<t_o)[/tex]
Conclusion
The rule of decision is this one:
[tex]p_v >\alpha[/tex] We fail to reject the null hypothesis at the significance level [tex]\alpha[/tex] assumed
[tex]p_v <\alpha[/tex] We reject the null hypothesis at the significance level [tex]\alpha[/tex] assumed
