Respuesta :
Answer:
Parameter tested: Population mean [tex]\mu[/tex]
We have enough evidence to conclude that the population mean for this case differs from 40.
Step-by-step explanation:
1) Notation and definitions
n= 28 represent the sample size
[tex]/bar x =7.26[/tex] represent the sample mean obtained
[tex]\sigma =3.72[/tex] represent the population standard deviation known
[tex]\mu_o =40[/tex] the value that we want to compare or test
2) Concepts and formulas to use
The system of hypothesis that we need to check for this case are
Null Hypothesis: [tex]\mu =40[/tex]
Alternative hypothesis: [tex]\mu \neq 40[/tex]
We assume that the sample mean follows a normal distribution.
When conduct a Z test, to analyze if the population mean is equal to a apecified value [tex]\mu_o =40[/tex]:
In order to check the hypothesis we need to use the following statistic
[tex]z=\frac{\bar X -\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
A one sample test of means "compares the mean of a sample to a prespecified value and tests for a deviation from that value".
Check for the assumptions that he sample must satisfy in order to apply the test
• The dependent variable must be continuous (interval/ratio). Satisfied
• The observations are independent of one another. We assume it.
• The dependent variable should be approximately normally distributed. Satisfied
• The dependent variable should not contain any outliers. We assume it.
3) Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{7.26 -40}{\frac{3.72}{\sqrt{28}}}=-46.57[/tex]
4) Statistical decision
95% of the values in the normal standard distribution are between -1.96 and 1.96, if we obtain a value of z=-46.57, the p value for a two tailed test would be almost 0. And for this case at any significance level [tex]\alpha[/tex] we will reject the null hypothesis that the population mean is 40. So we have enough evidence to conclude that the population mean for this case differs from 40.
