Consider the following hypothesis test H0 u 50 Ha u 50 A

Consider the following hypothesis test:

H_0: u = < 50

H_a: u > 50

A sample of 60 is used and the population standard deviation is 8. Use the critical value approach to state your conclusion for each of the following sample results. Use = .05

a. With x= 52.5, what is the value of the test statistic (to 2 decimals)?

Can it be concluded that the population mean is greater than 50?

b. With = 51, what is the value of the test statistic (to 2 decimals)?

Can it be concluded that the population mean is greater than 50?

c. With = 51.8, what is the value of the test statistic (to 2 decimals)? Can it be concluded that the population mean is greater than 50?

Can it be concluded that the population mean is greater than 50?

Solution

here H0: mu<=50 vs H1: mu>50

sample size=n=60    population SD=sigma=8 level of significance=0.05

then the test statistic is given as T=(xbar-50)*sqrt(60)/8 which under H0 follows a N(0,1) distribution.

and using critical value approach we reject H0 if t>tao_0.05 where t is the observed value of T and tao_0.05 is the upper 0.05 critical point of a N(0,1) distribution. now tao_0.05=1.64

a) xbar=52.5

then t=(52.5-50)*sqrt(60)/8=2.42 [it is the value of the test statistic]>tao_0.05=1.64

hence it can be concluded that the population mean is greater than 50 [answer]

b)xbar=51

then t=(51.5-50)*sqrt(60)/8=1.45 [it is the value of the test statistic]<tao_0.05=1.64

hence it can not be concluded that the population mean is greater than 50 as here the null hypothesis is not rejected [answer]

c) xbar=51.8

then t=(51.8-50)*sqrt(60)/8=1.74 [it is the value of the test statistic]>tao_0.05=1.64

hence it can be concluded that the population mean is greater than 50 [answer]