Question

Write a null and alternative hypothesis for single sample that tests the idea that the population mean is greater than or equals 25. Draw a sampling distribution which indicates the regions of rejection.

Answer 1

t-test for single Mean

suppose we wnt to test: 1. if a random sample of size n has been drawn from a normal population with a specified mean , say , or

2. if the sample mean differs significantly differs from the hypothetical value of the population mean

Under the null hypothesis , :

1. The sample has been drawn from the population with mean or

2. there is no significant difference between the sample mean ( x bar) and the population mean

so the test statistic is t /

where    

x bar and  

follows students t-distribution with (n-1) df

we can compare the calculated value of t with the tabulated value for some level of significance . if calculated > tabulated t, then null hypothesis will be rejected & if the calculated < tabulated t , will be accepted at that that level of significance

so Null Hypothesis , :

Alternative Hypothesis , :

as data given from the question

population mean = =25

sample mean = (x bar) not given in the question . let assume it to be 30

t = (30-25)/1.666    as n=1

the tabulated value of t for & 1 degree of freedom = 1 , where level of significance( let)

since t >

since calculated value of t is greater than the tabulated value

it is significant

hence is rejected at 5% level of significance

the graph of the sampling distribution along with rejection is given in the image