Question

"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!

Short answer

The alternate hypothesis states that the mean number of bubbles is at least less than$22$remains true.

Step-by-step solution

Given information

The average number of bubbles each time is at least$22$.

Sample size$=64$

Explanation

Hypothesis:

The null hypothesis claims that the mean number of bubbles is at least$22$

$$\begin{gathered} H_0:\mu \ge 22 \\ {H}_a:\mu <22 \end{gathered}$$

The random variable is $\overline{X}$, the mean number of bubbles.

By using students't-distribution we get,

$$\begin{gathered} t_n-1=t_{64}-1 \\ =t_{63} \end{gathered}$$

The sample mean $\overline{X}=20$

Calculate the Zvalue,

$$\begin{gathered} t=\frac{\overline{X}-\mu }{s/\sqrt{n}} \\ \text{whereμispopulationmean,nissamplesize} \\ z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}} \\ =\frac{20-22}{6/\sqrt{64}} \\ =\frac{-2}{6/8} \\ =\frac{-2}{0.75} \\ =-2.667 \end{gathered}$$