Question

You are given the following hypotheses:

H₀: μ = 30
H_a: μ ≠ 30

We know that the sample standard deviation is 10 and the sample size is 70. For what sample mean would the p-value be equal to 0.05? Assume that all conditions necessary for inference are satisfied.


The sample mean should be at most  or at least  (please round each answer to two decimal places)

Answer 1

Let X be a random variable and X~N(, ) where ,  both are unknown.

Here, the hypotheses are

H₀:=30 against H₁:30.

Let be the sample mean corresponding to the sample of size n from X.

Thus, ~N(, /n).

Under H₀, the test statistic is, t=√70(-30) /10, which follows t distribution with degrees of freedom (n-2)=68. Since the test is a both tailed test and at 5% level of significance the p value is given as 0.05,

P(l t l t_{0.025, 68}) =0.05

Or, P(l t l1.99) = 0.05[from the distribution table]

Or, P(l (-30)*√70/10 l1.99) =0.05

Or, P[(-30)2.38 or (-30) -2.38]=0.05

Or, P[32.38 or 27.62]=0.05

Thus, the sample mean should be at most 27.62 or at least 32.38 so that the p value be equal to 0.05.