I don't get it, please explain

Question

You wish to test the following claim (HaHa) at a significance level of α=0.10α=0.10.

      Ho:μ=51.8Ho:μ=51.8

      Ha:μ<51.8Ha:μ<51.8

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=7n=7 with mean M=36.3M=36.3 and a standard deviation of SD=13.2SD=13.2.

What is the test statistic for this sample? (Report answer accurate to three decimal places.)

test statistic = 

What is the p-value for this sample? (Report answer accurate to four decimal places.)

p-value = 

Answer 1

For a normal population with unknown standard deviation and an extremely small sample size (less than 10), go to a t-test. Take the number of degrees of freedom (df) as (7 − 1) or 6 and then take the Critical Value from the t-table at the intersection of df = 6 and α = 0.1 to obtain 1.439756.

For a "less-than" alternative hypothesis, one has a left-tailed test, so the critical value takes a negative sign and is written as -1.439756 or -1.4398.

Now calculate the test statistic t from t = (x-bar − µ₀) ÷ (s/√n) or (36.3 − 51.8) ÷ (13.2/√7) or -3.106753433 or -3.107.

With -3.106753433 (the test statistic) less than -1.439756 (the critical value), H₀ : µ = 51.8 is rejected at the "α = 0.1" level of significance.

|-3.106753433| or 3.106753433 falls between 2.44691 (under p = 0.025) and 3.14267 (under p = 0.010) for (df = 6) in the t-table so 0.0100 < p-value < 0.0250.