A weight loss program advertises "LOST 40 POUNDS IN 4 MONTHS".

Question

A random sample of n = 25 customers has x ¯= 31 pounds lost and s = 12. To contradict this claim test H0: μ = 40 versus H1: μ < 40 at α = 0.05.

 

 

Find the associated p-value. Round to four decimal places.

Answer 1

For n < 30, go to "Student's" t Distribution.

Take Degrees Of Freedom or df as (25 − 1) or 24.

Decide between:

H₀ : μ = 40

H₁ : μ < 40.

The Alternative Hypothesis H₁ : μ < 40 (as opposed to H₁ : μ ≠ 40)

dictates a left-tailed test instead of a two-tailed test.

Build the t-Test Statistic as T = (X_bar − μ)/S, here equal to (31 − 40)/12 or -0.75.

Accept (H₀ : μ = 40) if T is greater than -t_{(1 − 0.05)}. For df = 24, a t-Table intersecting

at t_0.95 and df = 24 gives T > -1.71.

For this problem, T at -0.75 exceeds -1.71, so (H₀ : μ = 40) is not rejected at Significance

Level α = 0.05.

To gain the P-Value, perform an interpolation with values from the t-Table as shown below:

-------------------------------t_{(1 − 0.8 = 0.2)}-------------P-Value-------------------t_{(1 − 0.75 = 0.25)}

(df = 24)-----------------------0.857---------------------0.75----------------------------0.685

Then analyze (0.75 − 0.857) ÷ (0.685 − 0.857) equals (P-Value − 0.2) ÷ (0.25 − 0.2)

which reduces to P-Value = 0.2311046512 equivalent to 0.2311.

The P-Value is the probability of obtaining a sample statistic as extreme as (or more extreme than)

the sample statistic gained if one assumes that the null hypothesis is true.

The P-Value is also defined as the smallest significance level at which the null hypothesis would be rejected.