Q:

A simple random sample of size 28 has mean x=7.26. g The population standard deviation is σ=3.72. The population is normally distributed. Can you conclude that the population mean differs from 40? Determine the parameter to be tested.

Accepted Solution

A:
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 definitionsn= 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 test2) Concepts and formulas to use  The system of hypothesis that we need to check for this case areNull 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.