Question

The makers of Excedrin Extra Strength advertise that its fast-acting formula delivers extra strength headache relief in no more than 15 minutes, on average. A consumer advocacy group believes that this claim is incorrect, and that the true average headache relief time is significantly longer than 15 minutes.

To test their belief, the consumer advocacy group took a random sample of 45 hospital patients who were currently experiencing a headache and administered a dose of Excedrin Extra Strength pain reliever. The amount of time it took for each patient to report relief from their headache, in minutes, was recorded. The sample mean headache relief time was 15.35 minutes, and the sample standard deviation was 0.95 minutes.

Using a significance level of .01, can the consumer advocacy group conclude that the true average headache relief time after taking Excedrin Extra Strength is longer than 15 minutes? Do a complete and appropriate hypothesis test.

Step 1 (Hypotheses)

H₀:    (Click to select)   π   x-bar   s   n   μ   σ   p      (Click to select)   =   ≠   ≤   >   ≥   <    

H_A:    (Click to select)   μ   x-bar   π   p   σ   s   n      (Click to select)   =   ≠   ≤   >   ≥   <    

Step 2 (Decision rule)

Using only the appropriate statistical table in your textbook, the critical value for rejecting H₀ is   (Click to select)   +   -   ±   . (report your answer to 3 decimal places, using conventional rounding rules)

Step 3 (Test statistic)

Using the sample data, the calculated value of the test statistic is   (Click to select)   +   -   ±   . (report your answer to 2 decimal places, using conventional rounding rules)

Step 4 (Evaluate the null hypothesis)

Should the null hypothesis be rejected?    (Click to select)   yes   no   

Step 5 (Practical conclusion)

Can the consumer advocacy group conclude that the true average headache relief time after taking Excedrin Extra Strength is longer than 15 minutes?   (Click to select)   yes   no   

Using only the appropriate statistical table in your textbook, what is the most accurate statement you can make about the numerical value of the p-value of this hypothesis test?       

Answer 1

Given that, n = 45, x-bar = 15.35 minutes, s = 0.95 minutes

Level of significance = α = 0.01

Step 1 (Hypotheses):

The hypothesis is always written with respect to the population measure. Since Excedrin Extra Strength is testing that on average how much is headache releif time, they are testing the ppopulation mean, Thus,

H₀: μ = 15 minutes

H_A: μ > 15 minutes (one-tailed)

Step 2 (Decision rule):

Since n = 45 > 30, we will use z-table. Our hypothesis is right-tailed, so at α = 0.01, the critical value of z is +2.33.

Thus, the critical value for rejecting H₀ is +2.33.

Step 3 (Test Statistic):

The calculated value of test statistic is given by

z = (x-bar) - μ_H0 / (s/√n)

Here, z = (15.35-15) / (0.95/√45) = 0.35/0.1416 = +2.47

Using the sample data, the calculated value of the test statistic is +2.47

Step 4 (Evaluate the null hypothesis):

Since the calculated z-value is greater than the critical z-value i.e. +2.47 > +2.33, we can Reject the null hypothesis at 1% level of significance.

So, the null hypothesis should be rejected. The answer is Yes.

Step 5 (Practical Solution):

Since null hypothesis is rejected, the consumer advocacy group can conclude that the true average headache relief time after taking Excedrin Extra Strength is longer than 15 minutes.The answer is Yes.

Using the z-table, the p-value for this hypothesis test is given by:

p-value = 1 - 0.9932 = 0.0068

We can reject the null hypothesis as p-value < 0.01.

(p-value is the least value of α at which the null hypothesis is just rejected).