Question

A) Set up the Hypothesis test

B) Determine the distribution needed

C) State what your random variable

D) State the distribution to use for the test

E) State the Test Statistic

F) Calculate the p-value using the normal distribution for proportions

G) In one to two complete sentences, explain what the p-value means for this problem

H) State about the Null Hypothesis

I) Draw and label the Graph for this problem

J) Compare α and the p-value

K) Conclusion

Riddle begins below:

"Blowing Bubbles," by Sondra Prull

Studying stats just made me tense,

I had to find some sane defense.

Some light and lifting simple play

To float my math anxiety away.

Blowing bubbles lifts me high

Takes my troubles to the sky.

POIK!

They're gone, with all my stress

Bubble therapy is the best.

The label said each time I blew

The average number of bubbles would be at least 22.

I blew and blew and this I found

From 64 blows, they all are round!

But the number of bubbles in 64 blows

Varied widely, this I know.

20 per blow became the mean

They deviated by 6, and not 16.

From counting bubbles, I sure did relax

But now I give to you your task.

Was 22 a reasonable guess?

Find the answer and pass this test!

Answer 1

A) Set up the Hypothesis test:

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:μ ≥22

Ha:μ <22

B) Determine the distribution needed: Z-test of normal distribution since sample size is large we estimate population standard deviation by sample standard deviation.

C) State what your random variable:sample mean is the random variable because its value depends on what the particular random sample happens to be.

D) State the distribution to use for the test:Z-test of normal distribution since sample size is large we estimate population standard deviation by sample standard deviation.

E) State the Test Statistic:

Z=​​~Z_0.05

t===-2.66~Z_0.05 (calculated)

the significance level is α=0.05, and the critical value for a left-tailed test is Zc​=−1.64.

F) Calculate the p-value using the normal distribution for proportions:

p-value is p=0.0038

G) In one to two complete sentences, explain what the p-value means for this problem:

The p-value is used as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected. A smaller p-value here=0.0038<0.05 means that there is stronger evidence in favor of the alternative hypothesis.

H) State about the Null Hypothesis:

Since it is observed that Z=−2.667<Zc​=−1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0038, and since ;p=0.0038<0.05, it is concluded that the null hypothesis is rejected.

I) Draw and label the Graph for this problem:

J) Compare α and the p-value:

p-value is p=0.0038, and since p=0.0038<0.05, it is concluded that the null hypothesis is rejected.

K) Conclusion:

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 22, at the 0.05 significance level.

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